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C
1.
Which of the following phenomena could be explained by classical physics and
did not require a quantum hypothesis in order to make theory agree with
experiment?
(a)
(b)
(c)
(d)
Blackbody spectra
(e)
Diffraction of light
(f)
Low-temperature heat capacity in (g)
perfect crystals
The photoelectric effect
(h)
2.
If a normalized wave packet Ψ is given as ! ( x, y, z,t ) =
Gravity
(b) and (e)
(a), (b), and (d)
None of the above
#
$ cn " n ( x, y, z)e %iE t / h ,
n
n =1
what is the probability that an experiment will cause the system to collapse to the
specific stationary state j?
(a)
|cj|2
(b)
(c)
(d)
Quantum mechanics does not allow (e)
you to know this probability
One
(f)
< ψj | H | ψj >
(g)
cj*
(h)
3.
Which of the following statements about the de Broglie wavelength λ are true?
(a)
λ=h/p
(b)
λ decreases as mass increases if (f)
velocity is constant
λ increases as momentum increases (g)
λ is a constant, like Planck's (h)
constant
(c)
(d)
(e)
cj
(b) and (d)
Only Schrödinger's cat knows
A particle that has zero velocity has
an infinite de Broglie wavelength
All of the above
(a), (b) and (e)
(c) and (d)
4.
Which of the below equations can be false for an arbitrary pair of orthonormal
functions f and g?
(a)
(b)
(c)
(d)
< |f|2 >< |g|2 > = 1
<f|H|g>=0
<f|g>=0
fg = 0
(e)
(f)
(g)
(h)
f*g – g*f = 0
(a) and (c)
(b), (d) and (e)
All of the above
NAME: ________________________________________________________________
2
5.
Which of the below expectation values are or may be non-zero?
(a)
(b)
(c)
< sinx | x | cosx >
< sin2x | x | cos2x >
< f | [A,B] | g > where A and B
commute
< Ψm | H | Ψn > where Ψm and Ψn
are non-degenerate stationary states
(d)
(e)
(f)
(g)
<f | g > – < g | f >*
<µ mn> for a forbidden transition
(a), (d) and (e)
(h)
(b), (c), and (f)
6.
Which of the following did Bohr assume in order to derive a model consistent
with the photoemission spectra of one-electron atoms?
(a)
The electron is a delocalized wave
(b)
(d)
The ionization potential is equal to (f)
the work function
The one-electron atom is like a (g)
particle in a box
The Coulomb potential is quantized (h)
7.
Which of the following statements about a well behaved wave function is true?
(a)
It must be continuous
(e)
(b)
It may take on complex values
(f)
(c)
(d)
It must be quadratically integrable
(g)
It must be equal to its complex (h)
conjugate
(c)
(e)
The angular momentum of the
electron is quantized
(a) and (b)
(b) and (d)
None of the above
Its square modulus has units of
probability density
It must be an eigenfunction of the
momentum operator
(d) and (f)
(a), (b), (c), and (e)
NAME: ________________________________________________________________
3
8.
Which of the following statements are false about the free particle?
(a)
Its
Schrödinger
equation
" h2 d 2
%
$!
! E ' ( (x ) = 0
2
# 2m dx
&
(b)
It may be regarded as having a (f)
wave function that is the
superposition of a left-moving
particle and a right-moving particle
Its energy levels are all non- (g)
negative
Its energy levels are quantized
(h)
(c)
(d)
is (e)
Valid wave functions include
where
! ( x ) = Aeikx + Be"ikx
2mE
k=
h
Valid wave functions include
! ( x ) = N coskx where N is a
normalization constant and k is
defined in (e) above
All of the above
None of the above
9.
Given a particle of mass m in a box of length L having the wave function
2
# n"x %
, what is the energy of the level corresponding to n = 4?
!( x ) =
sin$
L
L &
(a)
Since this wave function is not an
eigenfunction of the Hamiltonian
the question cannot be answered
< Ψ | px2 | Ψ >
16 times the energy of the ground
state
8" 2 h 2
mL2
(b)
(c)
(d)
! 10.
(e)
8h2 / mL2
(f)
(g)
(c) and (d)
(b) and (e)
(h)
None of the above
On which of the below functions does the parity operator Π act in the fashion
Π[f(x)] = (–1)f(x)?
(a)
x
(e)
(b)
(c)
(d)
x3
eix
sinx
(f)
(g)
(h)
Any
eigenfunction
Hamiltonian
(b) and (d)
(a), (b), and (d)
(b), (d), and (e)
of
the
NAME: ________________________________________________________________
4
Short-answer (20 points)
Prove that, given a pair of normalized but not orthogonal functions ψ1 and ψ2, the
function ψ3 = ψ2 – Sψ1 is orthogonal to ψ1 if S is the overlap integral of ψ1 and ψ2. Is ψ3
normalized? (Use the back of the page if necessary).
We evaluate
"1 "3 = "1 "2 # S"1
= "1 "2 # S "1 "1
= S # S •1
=0
which proves the orthogonality (we use normalization of ψ1 in going from line 2 to 3).
!
With respect to normalization, we evaluate
"3 "3 = "2 # S"1 "2 # S"1
= "2 "2 # 2S "1 "2 + S 2 "1 "1
= 1# 2S • S + S 2
= 1# S 2
so ψ3 is only normalized for the boring case of S = 0 (which means that ψ1 and ψ2 were
orthogonal to begin
with since no change to ψ2 is required to generate ψ3).
!
NAME: ________________________________________________________________