a small physics homeworke

Phys 301 Quantum Physics I Prof. M.M. Sharma

November 2, 2016

Problem Set 4

1. For any eigenfunction ψn of the infinite square well,

(a) Show that /2Lx  .

(b) Show that 2

22 2

)2(3 n

LL x  ,

where L is the size of the well.

2. Consider a system whose wavefunction at t = 0 is

(x), 20

7 x)(

20

3 x)(

5

1 x,0)(

321  

where x)( n  is the eigenfunction of the nth state of an infinite square well potential

of width a with the energy eigenvalues )2/( 2222

n manE  .

(a) Show that the wavefunction is normalized.

(b) Calculate the average energy of this system.

(c) Find the state t)x,( at any later time t and evaluate the average value of the

energy. Compare the result with the value obtained in (a). Does it depend upon

time and why ?

(d) What are the frequencies found in oscillations of the probability density and what is the periodicity of the oscillations, i.e., after what time interval does the

probability density return to its initial value?

3. The nuclear potential that binds protons and neutrons in the nucleus of an atom is often approximated by a square well. Imagine a proton confined in an infinite square

well of length 10-5 nm, a typical nuclear diameter. Calculate the wavelength and energy

associated with the photon that is emitted when a proton makes a transition from the

first excited state (n = 2) to the ground state (n = 1). In what region of the

electromagnetic spectrum does this wavelength belong ?

Formulas which can be useful:

a

axx

a

ax dxaxx

)cos()sin( )sin(

2  3

22

2

2 )cos()2()sin(2 )sin(

a

axxa

a

axx dxaxx

 