1
Quantum Mechanics (II): Homework 1
Due: March 9, 2013
Ex.1 Magnetic translation operator
Consider an electron in a uniform magnetic field H. Except for the magnetic field, the electron is otherwise free.
(a)5 Verify that the vector potential A(r) can be expressed as 21 H × r (this is the so-called symmetry gauge.).
(b)5 Let the wavefunction of the electron be ψ(r) which satisfies the Schrödinger equation Ĥ0 (r)ψ(r) = εψ(r). Under
a translation r → r + a, one expects that the system is translationally invariant. Show that however, in the symmetry
gauge, if a ⊥ H, Ĥ0 (r) is not invariant, i.e., Ĥ0 (r + a) 6= Ĥ0 (r). In fact, one finds that
e
e
Ĥ0 (r + a) = exp i
r · H × a Ĥ0 (r) exp −i
r·H×a .
2h̄c
2h̄c
(c) 10 If we define T̂a0 ψ(r) as the translated state of ψ(r) and require that T̂a0 ψ(r) satisfies the same Schrödinger
equation as ψ(r) does. Show that
e
r·H×a .
T̂a0 ψ(r) = ψ(r + a) exp −i
2h̄c
In other words,
e
T̂a0 = exp −i
r · H × a T̂a† ,
2h̄c
where T̂a is the ordinary translation operator. Thus, the probability density | T̂a0 ψ(r) |2 = | ψ(r + a) |2 is getting
translated.
(d) 10 Show that in general two magnetic translations operators, T̂a0 and T̂b0 , do not commute and they satisfy
e
T̂a0 T̂b0 = T̂b0 T̂a0 exp i H · a × b .
h̄c
(e) 10 We now put this electron in a two dimensional lattice with the primitive translation vectors c and d, and
assume that H =Hẑ. The Hamiltonian is
2
1
h̄
e
Ĥ(r) =
∇− A + V (r),
2m i
c
where V (r) is periodic: V (r + mc + nd) = V (r) for any integers m and n. Show that [T̂c0 (r), Ĥ(r)] = 0 and
[T̂d0 (r), Ĥ(r)] = 0 .
(f ) 10 Show that when the magnetic field is such that the flux through the primitive cell is p/q·hc/|e| (p and q are
two integers, which are relative prime, i.e., which have no common factors, and hc/2|e| is the so-called flux quantum),
0
0
0
0
[T̂nc
(r), T̂mqd
(r)] = 0 for any integers n and m. Hence the commutation relations between T̂nc
(r), T̂mqd
(r),and Ĥ(r)
are the same as those of T̂c (r), T̂d (r),and Ĥ0 (r) where Ĥ0 (r) is the Hamiltonian with zero magnetic field. The magnetic
field thus introduces new periodicity into the system of the electron.
Ex 2 10 (i) Suppose that ψ is known to be a simultaneous eigenstate of two Hermitian operators  and B̂ which
anticommute,
ÂB̂ + B̂Â = 0.
What can you say about the eigenvalues of  and B̂ for state ψ? Illustrate your point using the parity operator and
the momentum operator.
(ii) An example of anticommuting operators is the so-called chiral Hamiltonian. A Hamiltonian H is chiral if one can
find an operator Σ that anticommutes with H, i.e., {H, Σ} = 0. What can you say about the energy eigenvalues?
Ex. 3 Scale transformation and the squeezed state Consider a scale transformation Ŝ0 (ε) which is defined to
change the length scale in the following way
hx|Ŝ0 (ε)|ψi ≡ ψ(xe−ε ),
that is, x is scaled down by a factor exp(−ε).
(a) 5 The scale transformation Ŝ0 (ε) thus defined does not preserve the normalization of ψ. Show that if we define
2
Ŝ(ε) ≡ Ŝ0 (ε) exp(−ε/2), Ŝ(ε) preserves the normalization of ψ. The resulting operator Ŝ(ε) is the usual definition for
the scale transformation.
(b) 10 In terms of creation and annihilation operators, show that
Ŝ(ε) = exp ε/2(a†2 − a2 ) .
(c) 10 Show that Ŝ† (ε) x̂ Ŝ(ε) = eε x̂ and Ŝ† (ε) p̂ Ŝ(ε) = e−ε p̂. As a result the value of the product of uncertainties
∆x∆p is not changed. One can thus use Ŝ(ε) to squeeze the ∆x of any state, while not changing ∆x∆p !
(d) bonus problem Suppose the state of a harmonic oscillator (mass is m and its frequency is ω) at t = 0 is a
squeezed coherent state, i.e.,
|ψ(t = 0)i = Ŝ(ε) |λi,
where |λi is a normalized coherent state with eigenvalue λ. Find ∆x(t) and ∆p(t).
Ex. 4 Galilean transformation
(a) 5 Consider the Galilean transformation r → r − υt on a the state of a system, show that this transformation can
be realized by the operator
iυ
· (mr̂ − p̂t) .
Ĝ(υ, t) = exp
h̄
(b) 5 For a system of N particles, what would be the corresponding operator for the whole system?
The followings are bonus problems (+1):
(c) 15 To get a feeling what Ĝ(υ, t) does, consider two frames in one dimension: one with coordinate (x, t) is not
moving, the other with coordinate (x0 , t0 ) co-moves with the particle toward +x with the velocity υ. In other words,
the particle is at rest in the prime frame. Find the wavefunctions in both frames and show that they are related
exactly by Ĝ(υ, t). From the above, conclude that in general, consider two coordinates, (r, t) and (r0 , t0 ), that describe
the same space-time point in different frames, the wavefunctions in the two frames are related as follows
mυ 2
i
t) Ψ(r, t),
Ψ0 (r0 , t0 ) = exp − (mυ · r −
h̄
2
where r0 = r − υt and t0 = t. Note that any quantities with prime represent the quantites in the moving frame.
(d) 5 Argue that the Galilean invariance for Quantum Mechanics is expressed by the following condition
Ĝ† (υ, t) Û0 (t, t0 ) Ĝ(υ, t0 ) = Û(t, t0 ) × a phase factor,
where Û(t, t0 ) is the evolution operator.
(e) 10 Show that the above condition reduces to the form
0
∂
∂
†
Ĝ (υ, t) ih̄ − Ĥ Ĝ(υ, t) = ih̄ − Ĥ + f,
∂t
∂t
where f is an arbitrary real function of time. Verify that this condition is indeed satisfied by the Schrödinger
Hamiltonian. Find f in this case.
Ex. 5 10 Consider a rotation with respect to n̂ through an angle Φ. Suppose n̂ = (nx , ny , nz ), find the rotation
matrix in terms of nx , ny , nz ,and Φ.
Ex. 6 10 Consider a rotation specified by the Euler angles (π/4, π/4, π/4). What are the corresponding rotation axis
n̂ and Φ in the notations of Ex. 5? Use any mathematical software (Mathematica, Maple, ...) to help you solve this
problem.
Ex. 7 10 Show that if we restrict the 3-dimensional rotations to rotate only those lattice points in crystals, the trace
of the the rotational matrix must be an integer. From this result, show that two dimensional crystals can not have
5-fold rotational symmetry.
Ex. 8 10 Consider a vector field V(r) = (x + 2y)x̂ + xŷ. Let the coordinates be rotated by 45◦ counterclockwise,
i.e.,
1
1 1
r0 = √
r.
2 −1 1
After rotation, what is the vector field V0 (r0 ) ?