PHYS 237

Practice problems for the final

The final exam will cover the elements of quantum mechanics covered in the course (material discussed in Ch. 5, 6, 7). It is highly recommended to do these problems on your own with the book closed. The list of equations is provided on the last page. The number of questions below is larger than will be on the actual exam.

1. Consider a 2-dimensional square well of size a. (a) what are the energies and the wavefunctions of stationary states? (work carefully through this one, it is extremely important to understand how to build solutions of 2d and 3d problems when the solution of a 1d problem is known).

(b) If a particle in a state a

ya

x

ayxyx pipi 3sin2sin2)()(),( 32 == , what is the

uncertainty of the total energy of the particle? (Hint: if you end up having a serious integral, you are not doing this problem in the most efficient way)

(c) Compute the expectation value of zL in the state described in (b).

2. Consider a 3-dimensional harmonic oscillator. The Hamiltonian is

++

++

+= 22

222

222

2

21

2

21

2

21

2

zmm

pymm

pxm

m

pH zyx .

(a) What is the energy of the ground state? What is the wavefunction? (I think you had this one on the homework, please study this again!)

(b) What is the expectation value of px in the ground state?

(c) Compute the expectation value of potential energy in the ground state.

(d) What is the expectation value of the square of angular momentum 2Lr in the ground state (Hint: will it be easier to compute in spherical coordinates?).

3. In this problem, we will talk about hydrogen atom. (a) Consider a hydrogen atom in a ground state. A curious physicist measures the location of electron in the atom. What is the probability that he will measure the value rmeas between 2a and 2a+0.001a (Hint: you dont need a calculator here express your answer in terms of Borh radius a and dont worry about the numerical value; if you end up having an integral, note that the range of integration is very small, so the function under the integral is very close to a constant in this range).

(b) Consider a hydrogen atom in a stationary state with n=3. How many different linearly-independent wavefunctions have energy E3? Dont forget to include the electron spin into consideration.

(c) Hydrogen atom is in a state n=4, l=3, m=2. Is the transition from this state to the state with n=2 an allowed transition? Explain why or why not.

(d) Hydrogen atom is in a state n=6, l=3, m=-1. From this state, which transitions are allowed (the atom is emitting a photon).

(e) Hydrogen atom is in a state n=4, l=2, m=2. Compute the expectation value of the square of the z-projection of angular momentum 2zL . (f) Hydrogen atom is in a superposition of 2 stationary states:

( ) ( )( )hrhrr /exp)(/exp)(2

1),( 31,2,310,0,1 tiErtiErtr mlnmlm += ====== . Compute the

expectation values of 2Lr

and Lz at arbitrary t>0. (Hint: the wavefunctions of stationary states are normalized; also, use the fact that 0)()( 1,2,30,0,1 = ====== rrdV mlnmlm

rr )

4. Consider a 1d infinite square well of width a. Two identical particles of spin s=1/2 (each has mass m) are in the well. (a) what is the energy of the ground state? what is the wavefunction of the ground state of the system? (naturally, include both the coordinate and the spin parts).

(b) Now consider the 1st excited state. What are the possible wavefunctions for this 2-particle system? (Hint: you are supposed to get 2 possible answers for the coordinate part of the wavefunction)

(c) Finally, consider that each particle has a charge q. Taking into account the charge of the particles, which of the states you found in b is likely to have lower energy and why? (Hint: for which state the particles will be further apart on average?)

EQUATION SHEET

)()()()(2

;),(),()(),(

2 222

2

22

xExxVdx

xdmt

txitxxVx

tx

m =+

=+

hh

h

prLxVxm

Hxxx

ip ),(2

, , 2

22rrrh

h =+

==

=

2

222

2manEn

hpi= ,

=

==

mn

mndxxx

a

nx

ax

a

nmn for 1for 0)()( ,sin2)(

0

pi

+=

21

nEn h )()( ,)( 22

224/1

0 xHeAxem

x nx

m

nn

xm

=

=

hh

h

pi

dxxAxA )()( +

= dxx

xixp )()(

+

= h

( ) ( ) ( ) ( )2

, ,222222 h== xppppxxx

22

42 12 n

emkE en =h

, 2

2

mkea

h= ,

041pi

=k ( ) ( ) ( ) arimmlnlnlm ea

rePrAr /3100

1)( , cos,, ==pi

),,()1(),,( ),,,(),,(sin

1 sin

sin1

,

22

222

rllrLrmrL

LiL

nlmnlmnlmnlmz

z

+==

+

=

=

hr

h

hr

h

3/2,...) 1/2,(s Fermionsfor 0,1,...),(s Bosonsfor ),(),(),(),( ),1,2()2,1(

12122121 ==+==

ssrrssrr

( )

0cossin

)sin(cos1cos

)cos(sin1sin !

2!

2!!2

0

2

21

0

/

22

0

/1212

0

/2

2

2222

=

=+=

==

=

=

+

+

+

+

+

dxa

mx

a

nx

dxe axa

x(ax)a

(ax) dxx

axa

x(ax)a

(ax) dxx andxex

andxexa

n

ndxex

a

-

x

naxn

naxn

n

axn

pipi

pi

pi