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.
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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
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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:
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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⟩
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