Download Q.M3 Home work 9 Due date 3.1.15 1

Q.M3 Home work 9
Due date 3.1.15
1
a) Read in Sakurai chapter 3.8 1 ”Schwinger’s oscillator model of angular momentum ”
.
b) In the Schwinger representation, quantum mechanics spin is expressed in terms of two
creation and annihilation operators a− and a+ , in the form Ŝ + = a†+ a− , Ŝ − = (Ŝ + )†
and Ŝ z = 21 (a†+ a+ − a†− a− ).
Show that this definition is consistent with spin commutation relations [Ŝ + , Ŝ − ] = 2Ŝ z .
c) Using the creation and annihilation representation , show that:
(a†+ )S+m (a†− )S−m
p
p
|S, mi =
|0i
(S + m)! (S − m)!
is compatible with the definition of an eigenstate of the total spin operator Ŝ 2 and Ŝ z .
2
Often we need to calculate an integral of the form:
Z
dΩ Yl∗3 m3 (θ, φ)Yl2 m2 (θ, φ)Yl1 m1 (θ, φ)
(1)
This can be interpreted as the matrix element hl3 m3 |Ŷml22 |l1 , m1 i, where Ŷml22 is an irreducible tensor operator.
a) Use the Wigner Eckart theorem to determine the restriction on the quantum numbers
so that this integral does not vanish.
b) Given the ”addition rule” for Legendre polynomials:
X
Pl1 (µ)Pl2 (µ) =
hl3 0|l2 0l2 0i2 Pl3 (µ)
(2)
l3
1
3.9 in the new addition
1
Use the Wigner Eckart theorem to prove :
Z
(2l2 + 1)(2l1 + 1)
dΩ Yl∗3 m3 (θ, φ)Yl2 m2 (θ, φ)Yl1 m1 (θ, φ) =
hl3 0|l2 0l1 0ihl3 m3 |l2 m2 l1 m1 i
4π(2l3 + 1)
ˆ
Hint: consider hl3 0|Y0l2 |l1 0i
3
Find the eigenfunctions of the following equation:
i~
∂
−~2 ~ 2
∂
∂
ψ(x, y, z, t) =
[∇ ψ(x, y, z, t) − 2λi(y ψ(x, y, z, t) + x ψ(x, y, z, t)) − λ2 (x2 + y 2 )ψ(x, y, z, t)]
∂t
2m
∂x
∂y
(3)
Where λ is constant.
4
Coherent state2 of a one-dimensional simple harmonic oscillator is defined to be an
eigenstate of the (non-Hermitian) annihilation operator a:
a|λi = λ|λi
where λ is, in general, a complex number.
Prove that:
|λi = e−|λ|
2 /2
†
eλa |0i
is a normalized coherent state.
2
A coherent state is the specific quantum state of the quantum harmonic oscillator whose dynamics
most closely resembles the oscillating behaviour of a classical harmonic oscillator. Further, in contrast
to the energy eigenstates of the system, the time evolution of a coherent state is concentrated along
the classical trajectories. The quantum linear harmonic oscillator, and hence coherent states, arise
in the quantum theory of a wide range of physical systems. They occur in the quantum theory of
light and other Bosonic quantum field theories. A coherent state has a precisely defined phase but
undefined particle number, which it is in contrast to a state with fixed particle number, where the
phase is completely random.
2