Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 8
By Kit Matan
1. Delta function potential well and step function (20 points)
Consider a particle of mass m in the potential
V (x) = V0 (x) (x),
where V0
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 11
By Kit Matan
1. Hydrogen atom
Assume that nlm denotes an eigenfunction of the hydrogen atom with principal quantum
number n, and angular momentum quan
8.04 Quantum Physics
Lecture X
The Schrdinger equation
o
ih
h2 2
(x, t) =
(x, t) + V (x)(x, t)
t
2m 2 x2
(101)
The Schrdinger equation governs the motion of a particle in one dimension in a
o
potential V (x).
h2 2
ih (x, t) =
+ V (x) (x, t)
(102)
t
2
8.04 Quantum Physics
Lecture XII
After the energy measurement
After the energy measurement with outcome Ei , the particle will be in the energy
eigenstate ui , and all subsequent energy measurements will yield the energy Ei .
What is the energy before mea
8.04 Quantum Physics
Lecture XIII
p = p ^ h = dx i x h = dx  i x h = dx ( )  i  x x h = dx i x = p ,
(131) (132) (133) (134) (135) (136)
where again we have used integration by parts and the fact that vanishes at . Consequently, p = p , i.e. all
8.04 Quantum Physics
Lecture XI
Since V (x) = 0 in the region where the particle can be found, the zeropoint
energy must be purely kinetic in this case. We could have estimated it using the
Heisenberg uncertainty relation: Conning the particle to a regio
8.04 Quantum Physics
Lecture XV
Onedimensional potentials: potential step
Figure I: Potential step of height V0 . The particle is incident from the left with energy
E.
We analyze a time independent situation where a current of particles with a welldened
8.04 Quantum Physics
Lecture XVIII
1 2
multiplied by u0 (y) = e 2 y at innity. Consequently, we need the series to terminate,
which requires m = 2m + 1 for some m. Thus,
h
1
En =
n = h n +
HO energy levels
(181)
2
2
Quantized energy levels of a harmonic
8.04 Quantum Physics
Lecture XIV
Normalization of wavefunctions in free space
The momentum eigenstates in the position representation, up (x) dened by
pup (x) =
h
up (x) = pup (x),
i x
(141)
and given by
1
h
eipx/
(142)
2h
cannot be normalized in free
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2006
Solutions to Exam 2
1. Wavefunctions for piecewise constant p otential (20 points)
(a) The wavefunctions in regions I, II, III and IV:
Gex
I (x) =
for x a and
h
2 2
2m
II (x) =
8.04 Quantum Physics
Lecture XVI
The potential barrier: tunneling
Figure I: Tunneling through a potential barrier. Assume E < V0 (classically particle is reflected). Outside barrier solutions to the SE are u(x) = Aeikx + Beikx u(x) = Ce
ikx
for x < a, f
8.04 Quantum Physics
Lecture XVII
For the single potential we have exactly one bound state (symmetric state), for
the double potential we always have one symmetric bound state, and we may have
(depending on the potential strength) also an antisymmetric
8.04 Quantum Physics
Lecture XX
Angular momentum
The eigenequation associated with angular momentum reads
L2 Y (, ) = 2mr2 EL (r)Y (, ) = const Y (, )
where 2mr2 EL is the eigenvalue, and
2
2
2 = 2 + cot + 1
h
L
sin 2
2
(201)
(202)
Similar to the HO
8.04 Quantum Physics
Lecture XIX
We then have
1
a
H = H 0
a + a H 0
= H,
1
= h + a h0 0
a
2
3
= h 0
a
2
3
= h
1,
2
(191)
(192)
(193)
(194)
(195)
i.e.,

= a 0 is also an energy eigenstate, but with eigenenergy 3 h instead of 1 h
1
8.04 Quantum Physics
Lecture XXII
Radial equation for spherically symmetric potential
The SE in 3D in spherical coordinates is
h2 2
2
L2
+
(r) +
(r) + V (r)(r) = E(r)
2m r2 r r
2mr2
(221)
using the ansatz (r) = R(r)Y (, ), and inserting for the angular
8.04 Quantum Physics
Lecture XXIII
Last time
Radial equation for given angular momentum eigenstate Ylm (, ) with quantum num
ber l
2 2
2 2 l(l + 1)
+
+
+ V(r) Rnl (r) = Enl R(r)
(231)
2m r2 r r
2mr2
can be written in form of 1D SE with eective potentia
8.04 Quantum Physics
Lecture IV
Last time
Heisenberg uncertainty xpx
h
2
as diraction phenomenon
Fourier decomposition
1
(x) =
dk(k )eikx
2
1
h
=
dp(p)eipx/
2h
1
(p) = (k )
h
(41)
(42)
Today
how to calculate (k )
interpretation of (x) and
8.04 Quantum Physics
Lecture III
Last time we discussed
boundary between CM and QM set by resolution of measurement apparatus
in phase space
transition between CM and QM
Fermats principle of stationary time
Light takes the path where there is no rst
8.04 Quantum Physics
Lecture V
Last time
 (x)2 and (k )2 as probability densities in position and momentum space,
respectively. . .
Parsevals theorem: dx (x)2 = dk (k )2
Normalization: 1= dx (x)2 = dk (k )2
Measurement:
Measurement
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 9
By Kit Matan
1. Solving the Schrdinger equation in the momentum representation (20 points)
o
Consider the timeindependent Schrdinger equation for a pa
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 10
By Kit Matan
1. Operator relations (15 points)
(a) (5 points) Show that for any operator O = c1 a + c2, where a and are linear operators,
b
b
and c1 a
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 6
By Kit Matan
1. Practice with delta functions (10 points)
The Dirac delta function may be dened as
( x) =
for x = 0
0 for x = 0 .
such that
dx (x)f (
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 4
By Kit Matan
1. Xray production. (5 points)
Calculate the shortwavelength limit for Xrays produced by an electron acceleration voltage
of 30 kV. Is
Massachusetts Institute of Technology
Quantum Mechanics I (8.04) Spring 2005
Solutions to Problem Set 1
By Kit Matan
1. Photon energy scale I (20 points)
According to quantum mechanics, electromagnetic radiation of frequency can be regarded as
consisting
8.04 Quantum Physics
Lecture VII
Double slit: mathematical model of interference pat
tern and photon scattering
To develop some more insight into interference, and the correlations between quantum
system and (classical) apparatus that lie at the heart of