Quantum Mechanics I, Physics 3143: Assignment 1, Fall 2010: due 9/7 in
class
Notation: We use t to denote /t, etc.
1. (a) The (angular) frequency, > 0, and wavenumber, k , of light propagating in the
vacuum are related by = c|k |, where c is the speed of
Quantum Mechanics I, Physics 3143: Assignment 6, Fall 2009 due 10/29
1.(a) Show from the denition
sinKx
K
x
(x) = lim
that (cx) = (x)/|c| for c
.
(b) From the denition in part (a), show that
(x) =
and also that (x) =
1
2
1
2
eikx dk,
ikx
e
dk.
(c) Th
Physics 3143 Solution Set 8
Problem 1:
In view of Problem 5.1, we can correct for the motion of the nucleus
in hydrogen by simply replacing the electron mass with the reduced
mass.
(a) Find (to two significant digits) the percent error in the binding
ener
Physics 3143 Solution Set 7
Problem 1:
(a) Apply S- to 1 0\ (Equation 4.177), and confirm that you get
!
2 1 - 1\
Equation 4.177 gives us: 1 0\ = 1 H + L
!
H1L
H2L
S- = S- + S- , so
2
H1L
H2L
S- 1 0\ = HS- + S- L 1 H + L = 1 @HS- L + HS- L + HS- L + HS- L
Physics 3143 Solution Set 11
Problem 1:
Find the momentum-space wave function, FH p, tL, for a particle in the
ground state of the harmonic oscillator. What is the probability (to 2
significant digits) that a measurement of p on a particle in this state
w
Physics 3143 Solution Set 10
Problem 1:
The density of copper is 8.96 gm cm3 , and its atomic weight is 63.5
gm/mole.
(a) Calculate the Fermi energy for copper (Equation 5.43). Assume
q = 1, and give your answer in electron volts.
Equation 5.43 gives:
EF
Physics 3143 Solution Set 12
Problem 1:
Write explicitly the Sx, S y, and Sz matrices for S=3/2.
To solve for these three matrices, we look to the general angular momentum
relations:
Sz s, m\ = H mL s, m\
S+ s, m\ = I
S- s, m\ = I
!
!
!
sHs + 1L - mHm + 1
Physics 3143 Solution Set 6
Problem 1:
(a) Starting with the canonical commutation relations for position and
momentum (Equation 4.10), work out the following commutators:
@Lz , xD = i y, @Lz , yD = -i x, @Lz , zD = 0
@Lz , pxD = i p y, @Lz , p yD = -i px
Physics 3143 Solution Set 9
Problem 1:
(a) Suppose you put both electrons in a helium atom into the n=2
state; what would the energy of the emitted electron be?
Let us consider the energy of each electron:
2
Z E1
4E
E = = 41 = E1 = -13.6 eV
n2
So the tota
Physics 3143 Solution Set 3
Problem 1: A free particle has the initial wave function
YHx, 0L = A e-a x
2
where A and a are constants (a is real and positive).
(a) Normalize Y(x,0).
To normalize Y we must guarantee that
2
- Y x = 1
- A e-ax 2 x = - A2 e-2
Physics 3143 Solution Set 4
Problem 1:
(a) Show that the sum of two Hermitian operators is Hermitian.
An operator is Hermitian iff X a b\ = Xa b\ . So for the sum of two operators
Aand B to be Hermitian, it is both necessary and sufficient to show:
`
`
Z
Physics 3143 Solution Set 1
Problem 1: Ultraviolet light of wavelength 350 nm falls onto a
potassium surface. The maximum energy of the photo-electrons is 1.6
eV. What is the work function of potassium?
The work function is the energy needed to extract an
Physics 3143 Solution Set 5
Problem 1:
1
0
Use Equations 4.27, 4.28, and 4.32 to construct Y0 and Y2 .
H2 l+1L Hl-mL!
We start with equation 4.32: Ylm = "# ei m f Plm Hcos qL
#
4 pHl+mL!
1
0
Y0 = "# P0 Hcos qL
p 0
4
0
Next we use 4.27 to evaluate P0 Hc
Physics 3143 Solution Set 2
Problem 1: Consider the wave function YHx, tL = A e-l x e-i w t ,
where A, l, and w are positive real constants.
(a) Normalize Y
To normalize Y we must guarantee that
2
- Y x = 1
- A e
-l x e-i w t 2
x = - HA e-l x e-i w t L H
I. PROBLEM 1
a). The wave function is correctly normalized if
[+30 waomx. 0)da: = 1
OC
For \I/(x. 0) = (110(:r) + \I/1(:I:) + + one can write
x :x:
f \l/*(I,O)\II(17.O)d;r [Home + was) + tax) + was»
(910(1) + \l/1(3:) + \IJ3(:1:) + \I/5(1:))d:1:
Quantum Mechanics I, Physics 3143: Assignment 5, Fall 2009 due 10/13
1. This question illustrates the economy of eort in using algebraic, as opposed to analytic, methods for harmonic oscillator calculations. Evaluate the so-called coordinate matrix
elemen
Quantum Mechanics I, Physics 3143: Assignment 8, Fall 2010
1. (a) The spin-1/2 operators act on a vector space C2 of state vectors. Find the normalized
eigenvectors and eigenvalues of the Pauli matrices,
10
0 i
01
.
and z =
, y =
x =
0 1
i0
10
Clearl
Quantum Mechanics I, Physics 3143: Assignment 2, Fall 2010
1. To justify the energy eigenfunction expansion for the wave function of a particle in a
box x [0, a] it is useful to know that any square integrable function dened on the range
x [a, a] can be w
Quantum Mechanics I, Physics 3143: Assignment 3, Fall 2010
1. Griths problem 2.7 p. 39 ( Read examples 2.2 and 2.3 before you answer.)
2. Consider the Fourier transform of the wave function (x, t),
(p, t) =
dx ipx/
e
(x, t),
2
and the inverse Fourier tran
Quantum Mechanics I, Physics 3143: Assignment 4, Fall 2010
2
1. (a) Substitute (x, t) = Aex eit/2 into the time dependent Schrodinger equation for
a particle of mass m, and determine for which potential V (x) and which value of it is a
stationary state.
(
Quantum Mechanics I, Physics 3143: Assignment 5, Fall 2010
1. (a) Show that the wave function,
n
it/2
z (x, t) = N e
n=0
(zeit )
un (x),
n!
with z a complex number, satises the time-dependent Schrodinger equation i = H ,
t
for the harmonic oscillator whos
Quantum Mechanics I, Physics 3143: Assignment 6, Fall 2010
1. Consider scattering from a step potential
0, x < 0;
V (x) =
V0 , x 0.
Suppose that incoming particles from x = , with energy E , are incident on the interface.
Calculate the incident, reected
1
APPENDIX 1: COMPLEX NUMBERS AND PROBABILITY
A.
Complex numbers and wave interference
The number i satises i2 = 1. Let us write a general complex number in cartesian and polar forms as z = x + iy =
rei , where x = Re(z ) and y = Im(z ) are real numbers,
1
APPENDIX 2: BOUND STATES IN ONE DIMENSIONAL PROBLEMS
A.
Bound state energy levels are non-degenerate in one dimension.
Let us consider the hypothesis that there are two linearly independent eigenfunctions u1 and u2 corresponding to
the same energy eigen
1
ADDITIONAL COMMENTS ON HERMITIAN OPERATORS
The remarks below are not intended to be complete. Rather, they supplement the material given in the lecture of
October 5, 2010.
The theory of Hermitian[1] operators arises naturally in quantum theory. In the l
1
BACKGROUND QUANTUM PHYSICS 1900-1925
In these notes we briey discuss several of the important developments in quantum physics in the period 1900-1925
before the invention of quantum mechanics by Heisenberg and Schrodinger [1, 2].
Blackbody radiation (P