Physics 264L: What to Know and Some Review Questions for
Midterm 2
What you should know
The following are most of the key facts, concepts, and skills that you should know.
1. You should continue to know by memory the values of the fundamental constants ℏ, k, c, e, K, G, me ,
and mp to the nearest power of ten in SI units, and you should be able to carry out order-of-magnitude
estimates to the nearest power of ten quickly and without having to multiply or divide any digits.
2. You should understand and be able to explain in one to two paragraphs the key details of four experiments that established the particle nature of the photon and the wave nature of massive particles:
(a) the photoelectric effect which confirmed that E = hf .
(b) the Compton effect that established that photons act like point particles.
(c) the Davisson-Germer experiment, which was the first to confirm that electrons have wave-like
properties and the wavelength was precisely in accord with the de Broglie relation λ = p/h.
(d) the two-slit interference experiment carried out with a monoenergetic beam of electrons, which
established both the wave-propert of electrons according to λ = p/h and demonstrated directly
the probabilistic meaning of the wave function Ψ.
3. How to normalize a given function f (x) so that ⟨f |f ⟩ = 1.
4. How to determine if two wave functions are orthogonal to one another.
5. You should know how to calculate the phase speed vϕ and group speed vg of waves satisfying some
given dispersion relation ω = ω(k), and you should know how to derive heuristically a linear wave
equation consistent with a given dispersion relation.
6. For a piecewise constant potential V (x) that is finite everywhere, you should know how to use continuity
of the wave function ψ(x) and continuity of its spatial derivative dψ/dx to obtain equations that let
you solve the discrete bound-state energies En and calculate explicit wave function solutions for those
energies.
7. For a potential V (x) containing one or more Dirac delta functions, you should know how to use
continuity of the wave function and integration over small spatial regions centered on each delta function
to obtain equations that allow one to solve for the bound-state energy values.
8. You should know how to determine visually and qualitatively if some given transcendental equation
has one or more solutions. (A conceptual example: what is the maximum number of intersections a
circle can have with the sine function?
9. You need to know how to prove key results involving solutions of the one-dimensional Schrodinger
equations such as:
(a) the orthogonality of bound states belonging to different energy values.
(b) that the 1d time-dependent Schrodinger equation preserves probabilities (the area under the
probability density curve |Ψ|2 is always equal to 1 if it starts off being equal to 1.
(c) relations like ⟨p⟩ = md⟨x⟩/dt.
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All of these results involve using the definition of the integral dot product, the fact that a wave function
Ψ(t, x) satisfies the one-dimensional Schrodinger equation, and one or more integration by parts.
10. How to calculate averages such as
⟨x⟩,
⟨x2 ⟩,
⟨p⟩ = m
d⟨x⟩
,
dt
⟨p2 ⟩ = ⟨ Ψ |(−iℏ∂x )2 Ψ ⟩,
(1)
given some explicit wave function Ψ(t, x), and so how to calculate the standard deviations σx =
√
⟨x2 ⟩ − ⟨x⟩2 and similarly for σp .
11. How to draw qualitatively correct wave functions of bound states with different energy values for a a
variety of one-dimensional potential V (x), and how to draw qualitative potentials consistent with some
given wave function or some given probability density.
12. How to work with a Dirac delta function, both in an integral and when solving the Schrodinger equation
for a potential V that involves a delta function.
13. How to find the time-dependent solution Ψ(t, x) of the Schrodinger equation corresponding to some
initial state Ψ(t0 , x) by expanding the initial state Ψ(t0 , x) = Ψ0 (x) in terms of the energy states ψn :
Ψ0 (x) =
∞
∑
cn ψn (x)
where
cn = ⟨ ψn | Ψ0 ⟩,
(2)
n=1
which then leads to the time-dependent state
Ψ(t, x) =
∞
∑
cn e−i(En /ℏ)t ψn (x).
(3)
n=1
You need to know how to interpret the physical meaning of the coefficients cn = ⟨ ψn | Ψ0 ⟩, as “the
amplitude to transition from the state on the right to the state on the left” when an energy measurement
is carried out, and that |cn |2 is the probability of an energy measurement of a system in the state Ψ
yielding the value En .
∑∞
You should know how to prove that n=1 |cn |2 = 1 in Eq. (3).
You should know how to show that the average energy of the state Ψ(t, x) is given by
⟨E⟩ =
∞
∑
En |cn |2 .
(4)
n=1
14. You should understand the implications of “collapse of the wave function” when a measurement is
carried out: not only does an experiment of a certain observable (position, energy, momentum, etc)
yield a particular value of the observable, the measurement also changes (collapses) the state of the
system to a wave function corresponding to the particular observable. So a position measurement that
yields the value x0 collapses the wave function Ψ(t, x) to the state δ(x − x0 ), an energy measurement
with value En collapses the wave function to ψn (x), a momentum measurement that yields the value p0
collapses the wave function to ei(p/ℏ)x , and so on. The collapsed wave function then has a profound
effect on what happens when a new measurement of the same system is carried out.
15. You should understand how the fermionic nature of electrons in a many-electron system leads to the
stacking of electrons in successive energy levels, with at most two electrons per energy level. From
this, you should be able to predict what are the wavelengths (or frequencies) of absorption peaks of
a many-electron system with given energy levels. For a conjugated organic molecule with N carbon
atoms, you should be able to predict roughly where absorption peaks will occur.
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Practice problems
1. French and Taylor:
(a) 2-6, 2-8 .
(b) 3-1, 3-3, 3-8, 3-11, 3-15 parts (b), (d), and (e), 3-16 parts (b), (c), and (d), and especially 3-18 on
page 153.
2. If the brackets ⟨f (x)⟩ denotes the average of some function over an interval [a, b]:
⟨f (x)⟩ =
show that
3. For the wave function
1
b−a
∫
b
f (x) dx,
(5)
⟨
⟩ ⟨ ⟩
(f − ⟨f ⟩)2 = f 2 − ⟨f ⟩2 .
(6)
Ψ(t, x) = Ae−c|x| e−iωt ,
(7)
a
normalize Ψ and calculate the averages (expectation values) ⟨x⟩, ⟨p⟩, and ⟨x2 ⟩.
4. A particle of mass m is in a state described by the wave function
Ψ(t, x) = Ae−a[(mx
2
/ℏ)+it]
,
(8)
where A and a are real constants. Find the constant A so that Ψ is normalized, and determine what
is the potential V (x) that appears in the Schrodinger equation.
5. For the one-dimensional Schrodinger equation with potential V (x) with bound state Ψ(t, x), show that
the average energy ⟨E⟩ at time t is given by
⟨E⟩ = ⟨ Ψ | (iℏ∂t )Ψ ⟩
(9)
given that Ψ can be written in terms of all the bound-state stationary solutions ψn (x) as
Ψ(t, x) =
∞
∑
cn e−i(En /ℏ)t ψn (x).
(10)
n=1
6. What is the value of
∫∞
0
[cos(3x) + 2] δ(x − π) dx?
7. What are the wave functions ψn (x) and energies En for a particle of mass m that moves freely (V = 0)
in a one-dimensional circular tube of radius R? This corresponds roughly to a bead of mass m that
can move without friction on a ring of wire.
8. Consider a potential V (x) consisting of three delta functions:
[
]
V (x) = −α δ(x + L) + δ(x − L) + βδ(x),
(11)
consisting of two negative spikes at x = ±L and one positive spike at x = 0. Here α > 0 and β > 0.
(a) Draw the qualitative form of the wavefunction ψ(x) for the lowest energy bound state E1 and the
second lowest energy bound state E2 (assuming the latter exists).
(b) Give the mathematical description of the wave function ψ1 (x) for all regions of space.
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(c) Derive and give necessary and sufficient conditions to determine the wave function over all of
space.
(d) Derive and give a transcendental equation whose solutions determine the lowest energy E1 .
9. A particle in an infinite square well of width L has the initial state
Ψ(t = 0, x) = A sin3 (πx/L) .
(12)
(a) Calculate ⟨x⟩ as a function of time.
(b) At time t, what is the probability of an energy measurement giving the value E2 ?
(c) What is the average energy of this system at time t?
10. Consider the potential V (x) given by
{
V (x) =
αδ(x) for −L < x < L,
,
∞
for |x| > L
(13)
consisting of an infinitely tall well of width L with a delta-function peak at its center (so α > 0)
(a) Sketch the wave functions ψ1 (x) and ψ2 (x) of the two lowest energy states and describe how they
differ from the corresponding wavefunctions when the delta function is absent (α = 0).
(b) Solve explicitly the time-independent Schrodinger equation for the stationary states ψn (x) and
their corresponding energies En , and discuss how the wave functions and energies are affected by
the delta function.
11. For the wave dispersion relation ω = c1 k 2 + c2 k 4 ,
(a) What is the phase speed vϕ and group speed vg , and which is larger in the limit of small wavelengths and in the limit of long wavelengths?
(b) Derive a one-dimensional partial differential equation that wave functions u(t, x) consistent with
this dispersion relation must satisfy.
12. Find and give expressions for the approximate wavelengths λ1 and λ2 of the two lowest energy absorption peaks of a conjugated carbon molecule consisting of three carbon atoms.
The following pages contains some relevant questions taken from older physics GRE exams.
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