MATH 117 - ELEMENTS OF MATHEMATICS

- Bra–ket notation Question Bank - Set 3

Question 1

Step-by-step solution: 1. |0⟩represents the state vector corresponding to the

basis state |0⟩. 2. |1⟩represents the state vector corresponding to the basis state

|1⟩. 3. Given |ψ⟩= 3|0⟩−2i|1⟩, we can express this state in bra-ket notation as

⟨ψ|= 3⟨0|−2i⟨1|. 4. Therefore, the state |ψ⟩= 3|0⟩ − 2i|1⟩in bra-ket notation

is ⟨ψ|= 3⟨0| − 2i⟨1|.Question 1: Express the state |ψ⟩= 3|0⟩ − 2i|1⟩in

bra-ket notation.

Step-by-step solution: 1. |0⟩represents the state vector corre-

sponding to the basis state |0⟩. 2. |1⟩represents the state vector

corresponding to the basis state |1⟩. 3. Given |ψ⟩= 3|0⟩−2i|1⟩, we can

express this state in bra-ket notation as ⟨ψ|= 3⟨0|−2i⟨1|. 4. Therefore,

the state |ψ⟩= 3|0⟩ − 2i|1⟩in bra-ket notation is ⟨ψ|= 3⟨0| − 2i⟨1|.

Question 2

|a⟩=2

−i,|b⟩=1

3i

Step-by-step solution: 1. Write the vectors in the bra-ket notation:

|a⟩=2

−i= 2|0⟩ − i|1⟩

|b⟩=1

3i=|0⟩+ 3i|1⟩

2. Find the inner product of the vectors:

⟨a|b⟩= (2∗⟨0| − i∗⟨1|)(|0⟩+ 3i|1⟩)

= 2∗⟨0|0⟩+ 2∗3i⟨0|1⟩ − i∗⟨1|0⟩ − i∗3i⟨1|1⟩

3. Evaluate the inner product:

= 2∗·1+6i·0−0−(−i)·(3i)

1

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.Ques-

tion 2: Utilize the bra-ket notation to find the inner product of the

following vectors in complex vector space:

|a⟩=2

−i,|b⟩=1

3i

Step-by-step solution: 1. Write the vectors in the bra-ket notation:

|a⟩=2

−i= 2|0⟩ − i|1⟩

|b⟩=1

3i=|0⟩+ 3i|1⟩

2. Find the inner product of the vectors:

⟨a|b⟩= (2∗⟨0| − i∗⟨1|)(|0⟩+ 3i|1⟩)

= 2∗⟨0|0⟩+ 2∗3i⟨0|1⟩ − i∗⟨1|0⟩ − i∗3i⟨1|1⟩

3. Evaluate the inner product:

= 2∗·1+6i·0−0−(−i)·(3i)

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.

Question 3

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩Question 3: Convert the following

equation from the given expression in Dirac (bra–ket) notation:

2

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩

Question 4

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.Question

4:

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.

3

Question 5

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 5: Use Bra-ket notation to represent the following vectors:

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 6

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).Question 6:

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

4

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).

Question 7

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 7: Express the following vector in bra-ket notation:

v =

3

−4

5

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 8

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 8: Use Bra–ket notation to represent the vectors

A=

2

3

5

and

B=

1

4

6

.

5

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 9

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.Question 9: Explain

the concept of operator in Bra-ket notation.

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.

Question 10

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

6

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 10: Convert the following expression from bra-ket notation

to matrix form:

⟨ψ| · (λ|ϕ1⟩+µ|ϕ2⟩)

where λand µare complex numbers, and |ψ⟩,|ϕ1⟩, and |ϕ2⟩represent

quantum states.

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 11

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

7

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 11: Convert the following vectors from standard notation

to bra–ket notation:

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

8

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 12

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.Question

12: Use Bra-ket notation to represent the state |ψ⟩=1

√2|0⟩+i

√2|1⟩.

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.

9

Question 13

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 13: Represent the following vector in bra-ket notation:

v=

3

−1

2

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 14

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

10

command in LaTeX.Question 14: Express the following vectors in

bra-ket notation: a) ⟨v1|=

3

−2i

4

b) |v2⟩=

i

2

−5i

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

command in LaTeX.

Question 15

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.Question 15: In quantum mechanics, what does the bra-ket

notation represent and how is it used in calculations?

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

11

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.

Question 16

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

12

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.Question 16: Express the following vectors

in Bra–ket notation:

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

13

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.

Question 17

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.Question 17: Convert the

following expression to Bra-ket notation:

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.

Question 18

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

14

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.Question 18:

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.

15

Question 19

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 19: Express the following vectors in bra-ket notation: a)

3

−1

2

b)

−i

2i

4

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

16

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 20

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

ˆ

iis represented as |i⟩

ˆ

jis represented as |j⟩

ˆ

kis represented as |k⟩

Therefore, v1in Bra-ket notation is:

v1= 3|i⟩ − 2|j⟩+ 5|k⟩

b) To express v2=−4ˆ

i+ˆ

jin Bra-ket notation, we follow the same

process as above:

Therefore, v2in Bra-ket notation is:

v2=−4|i⟩+ 1|j⟩Question 20:

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

17

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.Ques-

tion 2: Utilize the bra-ket notation to find the inner product of the

following vectors in complex vector space:

|a⟩=2

−i,|b⟩=1

3i

Step-by-step solution: 1. Write the vectors in the bra-ket notation:

|a⟩=2

−i= 2|0⟩ − i|1⟩

|b⟩=1

3i=|0⟩+ 3i|1⟩

2. Find the inner product of the vectors:

⟨a|b⟩= (2∗⟨0| − i∗⟨1|)(|0⟩+ 3i|1⟩)

= 2∗⟨0|0⟩+ 2∗3i⟨0|1⟩ − i∗⟨1|0⟩ − i∗3i⟨1|1⟩

3. Evaluate the inner product:

= 2∗·1+6i·0−0−(−i)·(3i)

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.

Question 3

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩Question 3: Convert the following

equation from the given expression in Dirac (bra–ket) notation:

2

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩

Question 4

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.Question

4:

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.

3

Question 5

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 5: Use Bra-ket notation to represent the following vectors:

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 6

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).Question 6:

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

4

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).

Question 7

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 7: Express the following vector in bra-ket notation:

v =

3

−4

5

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 8

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 8: Use Bra–ket notation to represent the vectors

A=

2

3

5

and

B=

1

4

6

.

5

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 9

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.Question 9: Explain

the concept of operator in Bra-ket notation.

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.

Question 10

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

6

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 10: Convert the following expression from bra-ket notation

to matrix form:

⟨ψ| · (λ|ϕ1⟩+µ|ϕ2⟩)

where λand µare complex numbers, and |ψ⟩,|ϕ1⟩, and |ϕ2⟩represent

quantum states.

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 11

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

7

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 11: Convert the following vectors from standard notation

to bra–ket notation:

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

8

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 12

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.Question

12: Use Bra-ket notation to represent the state |ψ⟩=1

√2|0⟩+i

√2|1⟩.

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.

9

Question 13

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 13: Represent the following vector in bra-ket notation:

v=

3

−1

2

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 14

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

10

command in LaTeX.Question 14: Express the following vectors in

bra-ket notation: a) ⟨v1|=

3

−2i

4

b) |v2⟩=

i

2

−5i

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

command in LaTeX.

Question 15

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.Question 15: In quantum mechanics, what does the bra-ket

notation represent and how is it used in calculations?

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

11

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.

Question 16

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

12

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.Question 16: Express the following vectors

in Bra–ket notation:

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

13

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.

Question 17

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.Question 17: Convert the

following expression to Bra-ket notation:

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.

Question 18

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

14

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.Question 18:

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.

15

Question 19

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 19: Express the following vectors in bra-ket notation: a)

3

−1

2

b)

−i

2i

4

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

16

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 20

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

ˆ

iis represented as |i⟩

ˆ

jis represented as |j⟩

ˆ

kis represented as |k⟩

Therefore, v1in Bra-ket notation is:

v1= 3|i⟩ − 2|j⟩+ 5|k⟩

b) To express v2=−4ˆ

i+ˆ

jin Bra-ket notation, we follow the same

process as above:

Therefore, v2in Bra-ket notation is:

v2=−4|i⟩+ 1|j⟩Question 20:

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

17

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.Ques-

tion 2: Utilize the bra-ket notation to find the inner product of the

following vectors in complex vector space:

|a⟩=2

−i,|b⟩=1

3i

Step-by-step solution: 1. Write the vectors in the bra-ket notation:

|a⟩=2

−i= 2|0⟩ − i|1⟩

|b⟩=1

3i=|0⟩+ 3i|1⟩

2. Find the inner product of the vectors:

⟨a|b⟩= (2∗⟨0| − i∗⟨1|)(|0⟩+ 3i|1⟩)

= 2∗⟨0|0⟩+ 2∗3i⟨0|1⟩ − i∗⟨1|0⟩ − i∗3i⟨1|1⟩

3. Evaluate the inner product:

= 2∗·1+6i·0−0−(−i)·(3i)

= 2 −3i2

= 2 −3(−1)

= 2 + 3

= 5

Therefore, the inner product of the vectors |a⟩and |b⟩is 5.

Question 3

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩Question 3: Convert the following

equation from the given expression in Dirac (bra–ket) notation:

2

A= 2|a⟩+ 3|b⟩−|c⟩

Step-by-step solution: 1. The given equation can be represented

in Dirac (bra–ket) notation as: A= 2|a⟩+ 3|b⟩−|c⟩

2. In Dirac notation, the state |a⟩represents the vector corre-

sponding to the state ”a”.

3. Therefore, the converted expression in Dirac notation for the

given equation is: A= 2|a⟩+ 3|b⟩−|c⟩

Question 4

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.Question

4:

Express the vector |v⟩= 3|a⟩+ 2|b⟩ − |c⟩in bra-ket notation for the

given vectors |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1.

Step-by-step solution:

1. Given |v⟩= 3|a⟩+ 2|b⟩ − |c⟩, we need to substitute the given

vectors |a⟩,|b⟩, and |c⟩into the expression for |v⟩.

2. Substitute |a⟩=1

0,|b⟩=0

1, and |c⟩=1

1into |v⟩= 3|a⟩+ 2|b⟩−

|c⟩.

3. We get: |v⟩= 3 1

0+ 2 0

1−1

1

4. Simplify the expression: |v⟩=3

0+0

2−1

1|v⟩=3

0+−1

−1

|v⟩=2

−1

5. Therefore, the vector |v⟩in bra-ket notation is |v⟩=2

−1.

3

Question 5

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 5: Use Bra-ket notation to represent the following vectors:

(a) v = 2ˆ

i+ 3ˆ

j

(b) w =−ˆ

i−4ˆ

k

Step-by-step solutions:

(a) v = 2ˆ

i+ 3ˆ

j

Using Bra-ket notation, we can represent v as:

v = 2|ˆ

i⟩+ 3|ˆ

j⟩

(b) w =−ˆ

i−4ˆ

k

Using Bra-ket notation, we can represent w as:

w =−|ˆ

i⟩ − 4|ˆ

k⟩

Question 6

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).Question 6:

Express the following quantum state using Bra-Ket notation:

ψ=1

√2(|0⟩+|1⟩)

4

Step-by-step solution: We can express the given quantum state ψ

using Bra-Ket notation as follows:

ψ=1

√2(|0⟩+|1⟩)

Thus, the quantum state ψis represented in Bra-Ket notation as

1

√2(|0⟩+|1⟩).

Question 7

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 7: Express the following vector in bra-ket notation:

v =

3

−4

5

Step-by-step solution: The vector v can be expressed in bra-ket

notation as:

v = 31 −42 + 53

Therefore, the bra-ket notation for the vector v is:

v = 31 −42 + 53

Question 8

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 8: Use Bra–ket notation to represent the vectors

A=

2

3

5

and

B=

1

4

6

.

5

Step-by-step solution:

1. The bra–ket notation for a vector

Ais represented as ⟨A|=

235, and for vector

Bas ⟨B|=146.

Therefore, the bra–ket notation for vectors

Aand

Bare:

⟨A|=235

⟨B|=146

Question 9

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.Question 9: Explain

the concept of operator in Bra-ket notation.

Step-by-step solution:

1. In Bra-ket notation, an operator is a mathematical function

that acts on a quantum state represented by a ket (—

>

) to produce

another quantum state. 2. An operator is denoted by a symbol

enclosed within angle brackets, such as

<

A—. 3. When an operator

acts on a ket, it is written as

<

A—

>

. 4. The result of the operation is

a new quantum state denoted as another ket, for example, if operator

A acts on state —

>

, the result would be A—

>

. 5. Operators play a

crucial role in quantum mechanics by describing physical observables,

transformations, and evolution of quantum states.

Question 10

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

6

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 10: Convert the following expression from bra-ket notation

to matrix form:

⟨ψ| · (λ|ϕ1⟩+µ|ϕ2⟩)

where λand µare complex numbers, and |ψ⟩,|ϕ1⟩, and |ϕ2⟩represent

quantum states.

Step-by-step solution: 1. Distribute the scalar multiplication by λ

and µ:

λ⟨ψ|ϕ1⟩+µ⟨ψ|ϕ2⟩

2. Express the inner products ⟨ψ|ϕ1⟩and ⟨ψ|ϕ2⟩in matrix form:

Let |ψ⟩=a

b,|ϕ1⟩=c

d, and |ϕ2⟩=e

f. The matrix form of the

given expression is:

a bc

dλ+a be

fµ

3. Perform the matrix multiplications:

ac ad

bc bdλ+ae af

be bf µ

4. Sum the two matrices:

acλ +aeµ adλ +afµ

bcλ +beµ bdλ +bfµ

Question 11

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

7

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 11: Convert the following vectors from standard notation

to bra–ket notation:

v1=2

−3

v2=

1

0

4

v3=

−3

1

2

5

Step-by-step solutions: 1. For v1:

v1=2

−3= 20 −31

2. For v2:

v2=

1

0

4

= 0 + 01 + 42 = 0 + 42

3. For v3:

v3=

−3

1

2

5

=−30 + 1 + 22 + 53

8

Therefore, the bra–ket notation for the given vectors are:

v1= 20 −31

v2= 0 + 42

v3=−30 + 1 + 22 + 53

Question 12

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.Question

12: Use Bra-ket notation to represent the state |ψ⟩=1

√2|0⟩+i

√2|1⟩.

Step-by-step solution:

1. The given state |ψ⟩=1

√2|0⟩+i

√2|1⟩can be represented as a linear

combination of the basis states —0

>

and —1

>

in the following way:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

2. We can rewrite the state |ψ⟩in Bra-ket notation using the given

coefficients:

|ψ⟩=1

√21

0+i

√20

1

3. Simplifying the above expression, we get the state |ψ⟩in Bra-ket

notation as:

|ψ⟩=1

√2|0⟩+i

√2|1⟩

Therefore, the state |ψ⟩in Bra-ket notation is 1

√2|0⟩+i

√2|1⟩.

9

Question 13

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 13: Represent the following vector in bra-ket notation:

v=

3

−1

2

Step-by-step solution: 1. The given vector v=

3

−1

2

can be rep-

resented in bra-ket notation as:

v=

3

−1

2

2. Therefore, the representation of the vector vin bra-ket notation

is:

v=

3

−1

2

Question 14

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

10

command in LaTeX.Question 14: Express the following vectors in

bra-ket notation: a) ⟨v1|=

3

−2i

4

b) |v2⟩=

i

2

−5i

Step-by-step solutions: a) To express ⟨v1|in bra-ket notation, we

write ⟨v1|= (3,−2i, 4), where the vector components are separated by

commas.

b) To express |v2⟩in bra-ket notation, we write |v2⟩=

i

2

−5i

, where

the vector components are enclosed in a matrix using the ”pmatrix”

command in LaTeX.

Question 15

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.Question 15: In quantum mechanics, what does the bra-ket

notation represent and how is it used in calculations?

Step-by-step solution: The bra-ket notation, also known as Dirac

notation, is a notation used in quantum mechanics to represent vec-

tors, operators, and inner products.

1. Bra vector: The bra vector, denoted as ⟨ψ|, represents the

complex conjugate transpose of a ket vector |ψ⟩. It is used to represent

the row vector corresponding to the column vector ket.

11

2. Ket vector: The ket vector, denoted as |ψ⟩, represents a vector

in a complex vector space. It is used to represent the column vector

corresponding to the row vector bra.

3. Inner product: The inner product of two vectors |ψ⟩and |ϕ⟩is

denoted as ⟨ψ|ϕ⟩and represents the projection of one vector onto the

other.

4. Outer product: The outer product of two vectors |ψ⟩and |ϕ⟩is

denoted as |ψ⟩⟨ϕ|and represents the matrix formed by the product of

the ket and bra vectors.

5. Calculation: To perform calculations using bra-ket notation,

one can multiply ket vectors with bra vectors to form matrices, apply

operators represented as matrices, and compute inner products to

find probabilities or transitions between quantum states.

Therefore, the bra-ket notation provides a concise and powerful

way to represent and manipulate quantum mechanical states and op-

erators.

Question 16

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

12

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.Question 16: Express the following vectors

in Bra–ket notation:

a) 2

−3

b)

−5

1

4

c)

1

−1

1

0

—

Solution:

a) To express the vector 2

−3in Bra–ket notation, we use the

following representation:

|v⟩=2

−3= 2|0⟩ − 3|1⟩

b) For the vector

−5

1

4

, we can represent it in Bra–ket notation

as:

|v⟩=

−5

1

4

=−5|0⟩+|1⟩+ 4|2⟩

c) Finally, the vector

1

−1

1

0

can be written in Bra–ket notation as:

|v⟩=

1

−1

1

0

=|0⟩−|1⟩+|2⟩

13

Therefore, the expressions in Bra–ket notation for the given vec-

tors are provided above.

Question 17

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.Question 17: Convert the

following expression to Bra-ket notation:

⟨ψ|(A+B)|ϕ⟩

Step-by-step solution: 1. The expression can be written as the

inner product of ⟨ψ|and the operator (A+B)acting on the ket |ϕ⟩. 2.

The operator (A+B)acting on the ket |ϕ⟩is represented as (A+B)|ϕ⟩.

3. Therefore, the given expression in Bra-ket notation is:

⟨ψ|(A+B)|ϕ⟩

This represents the inner product of the Bra ⟨ψ|with the result of

the operator (A+B)acting on the ket |ϕ⟩.

Question 18

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

14

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.Question 18:

Consider the following vectors in the quantum mechanics Bra–ket

notation:

|ψ⟩= 3|a⟩ − 2|b⟩+|c⟩

|ϕ⟩=|a⟩+ 2|b⟩−|c⟩

a) Find the inner product ⟨ψ|ϕ⟩.

b) Determine whether the vectors |ψ⟩and |ϕ⟩are orthogonal.

—

Step-by-Step Solution:

a) To find the inner product ⟨ψ|ϕ⟩, we need to take the conjugate

transpose of one vector and multiply it by the other vector.

Given |ψ⟩= 3|a⟩ − 2|b⟩+|c⟩and |ϕ⟩=|a⟩+ 2|b⟩−|c⟩,

Taking the conjugate transpose of |ψ⟩gives ⟨ψ|= 3⟨a| − 2⟨b|+⟨c|.

Now, the inner product ⟨ψ|ϕ⟩is calculated as:

⟨ψ|ϕ⟩= (3⟨a| − 2⟨b|+⟨c|)(|a⟩+ 2|b⟩−|c⟩)

= 3⟨a|a⟩+ 6⟨a|b⟩ − 3⟨a|c⟩ − 2⟨b|a⟩ − 4⟨b|b⟩+ 2⟨b|c⟩+⟨c|a⟩+ 2⟨c|b⟩−⟨c|c⟩

Now, to evaluate the inner product, use the orthonormal proper-

ties of the basis vectors.

b) To determine if the vectors |ψ⟩and |ϕ⟩are orthogonal, we check

if their inner product is equal to zero.

If ⟨ψ|ϕ⟩= 0, then the vectors are orthogonal.

You can now proceed to simplify the expression and check if the

inner product is zero to determine orthogonality.

15

Question 19

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 19: Express the following vectors in bra-ket notation: a)

3

−1

2

b)

−i

2i

4

Step-by-step solutions: a) To express the vector

3

−1

2

in bra-ket

notation, we use the following notation:

ket

3

−1

2

=

3

−1

2

16

Therefore, the bra-ket notation for the vector

3

−1

2

is:

ket

3

−1

2

=

3

−1

2

b) To express the vector

−i

2i

4

in bra-ket notation, we use the

following notation:

ket

−i

2i

4

=

−i

2i

4

Therefore, the bra-ket notation for the vector

−i

2i

4

is:

ket

−i

2i

4

=

−i

2i

4

Question 20

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

ˆ

iis represented as |i⟩

ˆ

jis represented as |j⟩

ˆ

kis represented as |k⟩

Therefore, v1in Bra-ket notation is:

v1= 3|i⟩ − 2|j⟩+ 5|k⟩

b) To express v2=−4ˆ

i+ˆ

jin Bra-ket notation, we follow the same

process as above:

Therefore, v2in Bra-ket notation is:

v2=−4|i⟩+ 1|j⟩Question 20:

Express the following vectors in Bra-ket notation:

a) v1= 3ˆ

i−2ˆ

j+ 5ˆ

k

b) v2=−4ˆ

i+ˆ

j

Step-by-step solutions:

a) To express v1= 3ˆ

i−2ˆ

j+ 5ˆ

kin Bra-ket notation, we first write

each unit vector in the form of a ket as follows:

17

ˆ

iis represented as |i⟩

ˆ

jis represented as |j⟩

ˆ

kis represented as |k⟩

Therefore, v1in Bra-ket notation is:

v1= 3|i⟩ − 2|j⟩+ 5|k⟩

b) To express v2=−4ˆ

i+ˆ

jin Bra-ket notation, we follow the same

process as above:

Therefore, v2in Bra-ket notation is:

v2=−4|i⟩+ 1|j⟩

18