1
Quantum Mechanics (I): Homework 4
Due: December 2
Ex.1 10 Estimate the ground state energy of a two-electron atom with nuclear charge Z by using the uncertainty
relations.
Ex.2 In early development of quantum mechanics, Einstein proposed a way to measure which slit that a particle
would pass through in two-slit interference experiment. His proposed experiment is shown in FIG. 1. When the
particle passes through the first slit a, owning to the momentum conservation it transfers a downward momentum
to the screen s0 it it passes through the upper slit b and an upper momentum to the screen if it passes through the
lower slit c.
(a) 5 If the moment of the incident particle is perpendicular to the screen s0 and is p, estimate the momentum
transferred to screen s0 for particles passing through either b or c.
(b) 10 Using (a) and the uncertainty principle, show that when one can determine which slit the particle passes
through, the two-slit interference pattern on the screen s2 is washed out.
S0 S1 S2 b a d c p D FIG. 1.
Ex.3 10 Consider a particle of mass m that moves freely in one dimension. The initial wavefunction of that the
particle at t = 0 is
ψ(x, 0) = (π∆2 )−1/4 exp(ik0 x − x2 /2∆2 ).
For all possible values of ∆, find the minimum of the product of ∆t∆E, where ∆E is the uncertainty of energy and
can be considered as the characteristic time of the particle’s motion. Verify that ∆t∆E ≥ h̄/2 is
∆t = ∆x(t)/ d<x>
dt
satisfied.
Ex.4 10 The physical meaning of canonical momentum can be also understood by directly calculating the contribution
of momentum for the classical EM fields due to a point charge. Consider a point charge q being placed at the origin
~ generated by a spatially confined current distribution. Given that
in the magnetic field B
~ =
E
q~r
4π0 r3
~ = ∇×A
~ with ∇ · A
~ = 0, show that the fields momentum P~ = 0
and B
r
~ r).
momentum of the charge and field is given by m d~
dt + q A(~
R
~ ×B
~ = q A(0).
~
d3 xE
Therefore, the total
Ex.5 10 Let Π ≡ p − qA
c be the mechanical momentum. Show that [Πα, Πβ ] =
field and αβγ is the Levi-Civita tensor defined by
123 = 231 = 312 = 1,
132 = 213 = 321 = −1,
ih̄q
c αβγ Bγ
where B is the magnetic
2
。一
•
•
•
IIa
-
ID
、民、
月
、民、民
L
\
screen
FIG. 2.
and all other αβγ are zeros.
Ex.6 (a) 10 Given that H =
1
2m
2
p − qc A + qϕ , show that the probability current (flux) is given by
J=
1 ∗
(Πψ) ψ + ψ ∗ (Πψ) ,
2m
where Π is the mechanicalp
momentum.
(b) 5 If one writes ψ(r) = ρ(r) exp[iφ(r)], where ρ(r) is the probability density at r. Verify that
ρ(r)
qA(r)
J=
h̄∇φ(r) −
.
m
c
Ex.7 (a) 15 Using the result of Ex.3, show that in the Heisenberg picture
d2 x
dΠ
1 dx
dx
m 2 =
=q E+
×B−B×
,
dt
dt
2c dt
dt
which is the quantum-mechanical version of the Lorentz force.
(b) 10 Consider a uniforma magnetic field B = B ẑ. Find hxi and hyi and time t in terms of hxi0 , hyi0 , and hvα i0 .
α
Here vα ≡ dx
dt , α = x or y and hi0 is the average at time t = 0.
Ex.8 20 Consider a harmonic oscillator of mass m and natural frequency ω. If the position of the oscillator at t = 0
RT
is x1 and that at T is x2 , calculate the classical action Scl (x1 , x2 , T )(≡ 0 dtL).
Ex.9 10 A beam of atoms is collimated by being passed through a small circular aperture (diameter is a) as shown
in Fig.2. If we assume that initially, atoms only have x-component momentum p0 , by using the uncertainty relation,
find the minimum diameter D of the spot on the screen (in terms of p0 , a,and L) for all possible values of a.
Ex.10 Wigner function
In Quantum mechanics, one can not define the probability density P(P, R) for finding particles with position R and
momentum P. There is, however, a quantum analogy of the classical distribution function P(P, R). This is known
as the Wigner function:
Z
1
d3 r e−iP·r/h̄ ψ ∗ (R − r/2)ψ(R + r/2).
D(P, R) ≡
3
(2πh̄)
Unlike the classical distribution function, the function D(P, R) is not positive definite and may become negative. (a)
10 Show that
Z
d3 P D(P, R) = |ψ(R)|2 ,
Z
d3 R D(P, R) = |φ(P)|2 ,
3
R
3/2
where φ(P) ≡ d3 R/(2πh̄) ψ(R) exp(−iP · R/h̄). In other words, just as P(P, R) does, appropriate integrations
of D(P, R) can reproduce probability densities both in P and R spaces.
(b) 10 Show that if the particle is free, the way that D(P, R) evolves is the same as that of the classical distribution
function P(P, R).
Ex. 10 10 Fixed Energy Green’s function
For many processes, one needs to consider the propagation of the particle without changing its total energy (for
instance, in the elastic scattering process). Therefore, let us consider the Fourier transformation of the Green’s
function G(x, t; x0 ) ≡ θ(t)U (x, t; x0 0)
Z ∞
i
G(E; x, x0 ) ≡
dt e h̄ (E+iε) t G(x, t; x0 ).
−∞
Here in order to guarantee the integral over t is convergent, one change E into E + iε with ε → 0+ , i.e., ε is a small
positive number. G(E; x, x0 ) thus describes the propagation of the particle from x0 to x without changing its total
energy. Show that
Z
X
dx
ρ≡
Re G(E; x, x) =
δ(E − En ),
πh̄
n
where En are the energy levels of the system. In other words, the quantity ρ gives the density of state.
Ex.11 20 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. Consider a collection of such two-state systems. Suppose at t = 0, the percentage of systems in |1i is p0 , the
remaining systems are all in the state of |2i. Find the density matrix ρ(t) for t > 0.