Download Indiana University Physics P301: Modern Physics Review Problems

Indiana University
Physics P301: Modern Physics
Review Problems #3
Note: The final exam is on Wednesday May 4, 5:00-7:00 pm in class. It is comprehensive and
will cover lecture, handouts and homework material pertaining to HW #1-11. The following
problems are intended to review the material since the last exam.
1. Consider a square well having an infinite wall at x = 0 and a wall of height U0 at
x = L.
(a) For the case E < U0 , obtain solutions to the Schrodinger equation inside the
well (0 < x < L) and in the region beyond (x > L) that satisfy the appropriate
boundary conditions at x = 0 and at x = ∞.
(b) What are the boundary conditions at x = 0 and x = L? Enforce these conditions
to find the allowed energies of the system.
(c) Are there conditions for which no solution is possible? Explain.
2. You are putting the electrical wiring in your new house, and you are considering using
aluminum wiring, which is a cheap and good conductor. However, you also know
that aluminum tends to form an oxide surface layer which can be as much as several
nanometers thick. Your requirement is that your transmission coefficient across any
contact must be T > 10−10 , or else the resistance will be too high for the high currents
you are using, causing a fire risk. Should you use aluminum wiring or not? Recall
that the transmission coefficient for tunneling through a square barrier of width w and
height U0 for a particle with energy E < U0 is given by
T ≡
ψ 2
T
ψI U02 sinh2 (κw)
= 1+
4E (U0 − E)
!−1
where κ2 = 2m(U0 − E)/h̄2 , which in the weak tunneling limit κw 1 can be approximated as:
16E(U0 − E) −2κw
e
U02
The oxide layer presents a roughly 10 eV barrier to the flow of electrons, which can be
T ≈
approximated to have energy E = 11 eV.
3. The wave function for quantum harmonic oscillator of mass m in the n = 2 state is
2
given by A 1 − 2 xb2 e−x
and b =
2 /2b2
. The potential energy is U (x) = 21 kx2 , where ωc2 ≡ k/m,
q
h̄/mωc . Find the energy of the quantum oscillator in this state, and sketch
the wave function.
1
4. Consider the angular function, h(θ, φ) = A sin θ sin φ, where A is a real constant. (This
is an eigenfunction of the operator L̂2 for a system with spherical symmetry, i.e., in
a central potential, U = U (r)). It has a total squared angular momentum eigenvalue
h̄2 l(l + 1), with l = 1.
(a) Show that h(θ, φ) is not an eigenfunction of the Lz operator −ih̄(∂/∂φ).
(b) What is the average value of measurements of Lz for systems in which h(θ, φ) is
the angular part of a wave-function for a particle in a central potential?
(c) Does the total angular momentum of the particle in such a state have a definite
value? If so, what it is? If not, why not?
5. The ground state wave function for the hydrogen atom is
s
ψ100 (r, θ, φ) = Y00 (θ, φ) × R10 (r) =
1
1
×2
4π
a0
3/2
e−r/a0 ,
where a0 is the Bohr radius.
(a) Write down, but do not evaluate, an integral for the probability of finding the
electron in the region 0 < r < a0 , noting that the expressions for R(r) and
Y (θ, φ) above are separately normalized. Recall that the element of volume in
spherical polar coordinates is given by dV = r2 sin θ dθ dφ dr where θ is the polar
angle (0 < θ < π) and φ is the azimuthal angle (0 < φ < 2π).
(b) What is the most likely value for the radius r at which the electron can be found
(i.e., the value that maximizes the probability)?
(c) Evaluate the probability that the electron can be found in the angular region
0 < θ < π/4. [You can do the integral just over θ if you want, but be sure to also
evaluate the normalization integral in this case.]
(d) Moving from hydrogen to helium, can both electrons in the ground state of the
atom have the same spatial wave function? How will the value of a0 , given by the
Bohr radius for hydrogen, change in the spatial wave function for helium?
6. The radial part Rnl (r) of the hydrogen atom wave function is specified by the following
differential equation, obtained from the 3-D Schrödinger Equation by separation of
variables:
h̄2 d2 (rRnl (r))
−e2
h̄2 l(l + 1)
+
+
rRnl (r) = En rRnl (r) .
−
2me
dr2
4πε0 r
2me r2
"
#
2
(a) From the expression above, describe how the behavior at r = 0 of the solutions to
this equation differs for s and p orbitals? Explain your answer. (Recall that the
sequence for the angular momentum quantum number in spectroscopic notation
is {s, p, d, f, g, . . .}.)
(b) For the solution for the radial part of the 3s state of hydrogen, R30 (r) = C (27 −
18r/a+2r2 /a2 )e−r/3a , where C and a are real constants (a being the Bohr radius),
write down (but do not evaluate) an expression that allows one to determine the
value of the constant C.
(c) Are there finite values of r at which it is impossible to find the electron in the 3s
state discussed in (b) above? If so, how many are there and what are they? If
not, explain why not.
3
Possibly useful formulas
Planck’s Constant, electron mass and charge, etc.
h = 4.14 × 10−15 eV · s = 6.63 × 10−34 J · s ;
me = 0.511 M eV /c2 = 9.11×10−31 kg ;
h̄c = 197 eV · nm ;
h̄ = h/(2π) ;
e = 1.60×10−19 C ;
1/(4πε0 ) = 8.99×109 N ·m2 /C 2 .
Rydberg-Ritz, Bohr Radius, etc. for Hydrogen:
1
= (Ei − Ef )/hc = Ry
λf i
1
1
− 2
2
nf
ni
e2
= −
8πε0 hc
!
Rydberg-Ritz Formula
1
1
−
ri rf
!
,
energy levels of Bohr atom
where rn = n2 a0 , with a0 = λC /2πα = 0.0529 nm, λC = h/(me c), and α ≡ e2 /(4πε0 h̄c) =
1/137.
Sinusoidal Waves:
v = λf,
k = 2π/λ
ω = 2πf = 2π/T .
de Broglie Relations:
E = hf = h̄ω
p = h/λ = h̄k .
QM Operators (position representation):
x̂ = x ,
Ê = ih̄
2
2 ∂
2
ˆ
p = −h̄
,
∂x2
h̄2 ∂ 2
Ĥ = pˆ2 /2m + U (x̂) = −
+ U (x)
2m ∂x2
∂
p̂ = −ih̄ ,
∂x
∂
,
∂t
Expectation Values:
hQi =
Z ∞
Ψ∗ (x, t)Q̂Ψ(x, t) dx
x=−∞
Schrödinger Equation (Time-Dependent):
ih̄
∂Ψ(x, t)
h̄2 ∂ 2 Ψ(x, t)
= Ĥ Ψ(x, t) = −
+ U (x) Ψ(x, t)
∂t
2m ∂x2
Schrödinger Equation (Time-Independent):
h̄2 d2 ψ(x)
E ψ(x) = Ĥ ψ(x) = −
+ U (x) ψ(x) ,
2m dx2
4
where Ψ(x, t) = ψ(x) e−iEt/h̄
Possibly useful formulas, cont’d
Heisenberg Uncertainty Relations:
δx · δp ≥ h̄/2
δE · δt ≥ h̄/2 ,
where δq ≡ (hq 2 i − hqi2 )1/2
Solutions to the 1-D infinite well potential (U = 0 from x = 0 to x = L):
s
ψn (x) =
nπx
2
sin
L
L
.
Normalization condition for Schrödinger Eqn. w/ Central Potential:
Z ∞ Z π Z 2π
r=0
θ=0
φ=0
∗
∗
(r)Rnl (r)Ylm
(θ, φ)Ylm (θ, φ) r2 sin θ dr dθ dφ = 1
Rnl
Schrödinger Equation for Hydrogen atom after separation of variables:
(ignoring spin-orbit, spin-spin, interactions)
d2 Φm (φ)
+ m2 Φm (φ) = 0
2
dφ
d2 Plm (θ)
1 dPlm (θ)
m2
Plm (θ) = 0
+
+
l(l
+
1)
−
dθ2
tan θ dθ
sin2 θ
#
"
h̄2 d2 (rRnl (r))
h̄2 l(l + 1)
−e2
−
+
+
rRnl (r) = En rRnl (r)
2me
dr2
4πε0 r
2me r2
"
#
Angular momentum operators
c = −ih̄ ∂ ,
L
z
∂φ
"
2
c2 = −h̄
L
∂2
1 ∂
1
∂2
+
+
∂θ2 tan θ ∂θ sin2 θ ∂φ2
#
Properties of Spherical Harmonics
c Y (θ, φ) = h̄m Y (θ, φ) ,
L
z lm
lm
c2 Y (θ, φ) = h̄2 l(l + 1) Y (θ, φ)
L
lm
lm
Energy Eigenvalues for the Hydrogen atom
En = −
e2
8πε0 a0
where a0 = 4πε0 h̄2 /(me e2 ) is the Bohr radius.
5
1
n2
,