1
Quantum Mechanics (I): Homework 3
Due: November 11
Ex.1 (a) 10 Show that if  has no explicit time dependence and [Ĥ, Â] = 0, then < Ân >, n = 1, 2, · · · are timeindependent constants. Here H is the Hamiltonian.
(b) 10 Suppose < Ân > in (a) are known and the eigenvalues of  are real and continuous (denoted by a). Devise a
method to calculate the probability density of finding a.
Ex.2 As mentioned in the class, the WKB approximation is not valid near the turning point x0 defined by E = V (x0 ).
If we expand V (x) around x0 and keep only to the linear term, we may write V (x) = E + A(x − x0 ) where A = V 0 (x0 )
. Let us consider the turning point where A > 0 .
(a) 5 Change the variable to y = (2mA/h̄2 )1/3 (x − x0 ), find the differential equation that ψ(y) satisfies.
(b) 10 The solution to the above equation is the Airy function
Ai(y). The asymptotic form of large negative y for
1
2
3/2
the Airy function is Ai(y) ∼ √π(−y)
+ π4 . Show that this implies that for x < x0
1/4 sin 3 (−y)
ψ(x) ∼
Z x
1
π
0
0
p
cos
p(x )dx +
·
h̄ x0
4
p(x)
1
Ex.3 Consider a two-state system characterized by the Hamiltonian
H = ∆ [2|1 >< 1| − |2 >< 2| + 2(|1 >< 2| + |2 >< 1|)] ,
where ∆ is a real number with the dimension of energy. |1 > and |2 > are normalized and are also orthogonal to each
other.
(a) 5 If one measures the energy of this system, what possible values one would get?
(b)10 Suppose the ket of the system is |ψ(t) >= α(t)|1 > +β(t)|2 >, what are the differential equations that α(t)
and β(t) obey?
(c) 10 If at t = 0, |ψ(0) >= |1 >, what is the probability of finding the system at |2 > at t > 0?
Ex.4 Quark potential
a wave
function ψE (x) that satisfies the one dimensional time-independent
Consider
h̄2 d2
Schrödinger equation − 2m dx2 + V (x) ψE (x) = EψE (x), where V (x) is a linear potential (which is known
reasonably good for describing quarks near each other)
V (x) = Cx with C > 0, for x > 0
= ∞ for x < 0 .
(a) 10 Let the Fourier transformation of ψE x be ΦE (p). Show that
3
i
p
− Ep
ΦE (p) = Aexp
Ch̄ 6m
where A is an integration constant.
R∞
(b) 10 Using the normalization condition for ΦE (p): −∞ Φ∗E (p)ΦE 0 (p)dp = δ(E − E 0 ), find A (assuming A is real).
(c) 5 The boundary condition for ψE (x) is ψE (0) = 0, find the equation that the energy eigenvalue E satisfies.
(d) 10 From this equation, one can solve for the discrete energy spectrum. The ground state energy is E0 =
2 2 1/3
2.338 C2mh̄
. Use WKB method to get approximated ground state energy and compare it with this value.
Ex 5 10 The evolution operator Û(t, t0 ) for a system described by the Hamiltonian
Ĥ =
p̂2
+ V (x̂)
2m
0
is given by Û(t, t0 ) = exp[−i/h̄ Ĥ(t − t )]. If we define G+ (xt; x0 t0 ) to be hx|Û(t, t0 )|x0 i · θ(t − t0 ), show that G satisfies
the differential equation:
2 2
∂
−h̄ ∂
ih̄ G+ −
+
V
(x)
G+ = ih̄δ(x − x0 )δ(t − t0 ),
∂t
2m ∂x2
in other words, G+ is a Green’s function of the Schrödinger equation. Note that without imposing appropriate initial
condition, the above equation also admits solutions that do not vanish when t < t0 . For this reason, this equation has
2
to be supplemented with the condition G+ (xt; x0 t0 ) = 0 for t < t0 .
Ex 6
(a) 10 Consider a particle of mass m incident on a one-dimensional step function potential V (x) = V θ(x) with energy
E > V . Calculate the reflection and transmission coefficients when the particle is incident from x = −∞.
(b) 10 Find the transfer matrix in problem (a). Using the formalism of transfer matrix to calculate the transmission
coefficient for a particle of mass m incident on the following potential with E > 0:
V (x) = 0, for |x| > a
V (x) = −V , for − a ≤ x ≤ a.
(c) Bonus (+1) Following Ex. 14 in Homework 1, consider a particle that is confined to move on the one-dimensional
lattice: x = na with n be integers and a being a positive constant. If the wavefunction at site x = na is denoted by
Ψn , for the step potential on the lattice, Ψn of fixed energy E satisfies
−tΨn−1 − tΨn+1 + Vn Ψn = EΨn ,
where t is a positive constant, Vn = 0 for n < 0 and Vn = V > 2t for n ≥ 0. Find the reflection and transimission
coefficients when the particle is incident from n = −∞ with energy E > V − 2t.
Ex 7 (a)15 Construct the S-matrix for the potential V (x) = −Aδ(x) . From the S-matrix, find the energies for the
bound states.
(b) Bonus (+1) Following (a), let us denote the bound state by φb (x) and the scattering states by
φk,R (x) → e−ikx + R eikx for x → ∞
T e−ikx
for x → −∞
φk,L (x) → eikx + R e−ikx for x → −∞
T eikx
for x → ∞.
Show that the following completeness relation is satisfied.
Z
X
1 X ∞
∗ 0
φb (x)φb (x ) +
dkφk,s (x)φ∗k,s (x0 ) = δ(x − x0 ).
2π s 0
b
Ex.8 10 Consider a particle of mass m in an asymmetric potential well given by V (x) = V2 if x < 0, V (x) = 0 if
0 < x < a, and V (x) = V1 if x > a. Find the condition when there is no bound state.
Ex.9
(a) 10 Consider a particle of mass m incident with energy E from x = −∞ upon a potential barrier V (x) shown in
FIG. 1. The asymptotic behaviors of V (x) are: V (x) → 0 when x a and x b. Find the transmission coefficient
by using the WKB approximation. Ｅ x a b FIG. 1.
(b) 10 Splitting of energy due to the potential barrier
Consider a particle of mass m in a potential U (x) shown in FIG. 2. U (x) consists of two symmetrical potential
wells. Let ψ0 (x)
R ∞ being the energy eigenstate of one of the energies E0 < U0 to the right potential well with the
normalization 0 ψ02 (x) = 1. Due to the possibility of tunneling, the particle will also occupy the left potential well.
As a result, the energy is splitted into E1 and E2 as shown in FIG. 2. Show that E2 − E1 = (2h̄2 /m)ψ0 (0)ψ00 (0).
By usingp
the WKB approximation with connection formula, express E2 − E1 in terms of an apppropriate integral of
|p(x)| = 2m(U (x) − E0 ) and the frequency ω of the classical motion of the energy E0 .
3
FIG. 2.
Ex.10 Suppose that the wavefunction of a particle at t = 0 is given by
ψ(x, 0) = (π∆2 )−1/4 exp(−x2 /2∆2 ).
The particle is free for t ≥ 0.
(a) 7 Find the position operator in the Heisenberg picture x̂H (t) ( for t > 0) in terms of the operators x̂ and p̂
defined in the Schrodinger’s picture. Calculate the commutator [x̂H (t), x̂H (t0 )].
(b) 8 Using results from (a), calculate ∆x(t) for t > 0.
Ex.11 Bonus problem (+1) Combining S-matrices coherently
(a) Consider two one-dimension local potentials described by the scattering matrices
r1 t01
r2 t02
S1 =
,
S
=
.
2
t1 r10
t2 r20
By combinging S-matrices appropriately, show that the total transmission coefficient is
T =
T1 T2
√
,
1 − 2 R1 R2 cos θ + R1 R2
where Ti = |ti |2 , Ri = |ri |2 , and θ is the phase shift acquired in one round-trip between two potentials.
(b) Apply the above formulation to find the total transmission coefficient as function as the total incident energy E
for the potential
V (x) = U0 (δ(x) + δ(x − d)).
Sketch the plot the total transmission coefficient versus energy E and show how to find the resonant energy when
T = 1.
(c) Argue that if instead, we combine the probabilities directly for two potentials, we would get the so-called incoherent
total transmission coefficient
T =
in this case the variable θ disappears.
T1 T2
,
1 − R1 R2