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Transcript
Physics 880K20: Problem Set 4
Due Wednesday, February 22 by 5PM
1. This problem applies to any system with harmonic oscillator modes,
including real harmonic oscillators and photons, among others.
Consider the Hamiltonian h̄ωa† a, where a† and a are the photon creation and annihilation operators. a† and a satisfy the commutation
relations [a, a† ] = 1.
(a). Let |ni be an eigenstate of this Hamiltonian, with eigenvalue En .
By considering the commutator [H, a†n ], show that a† |ni is an eigenstate
of H with energy En +√h̄ω. Show that if |ni is normalized, then the
state |n √
+ 1i = a† |ni/ n + 1 is also normalized. Hence, show that
†
a |ni = n + 1|n + 1i.
√
(b) Using an analogous method, show that a|ni = n|n − 1i.
(c). Show that |ni =
† n
(a
√ ) |0i,
n!
where |0i is the normalized ground state.
³ √
´
(d). For a harmonic oscillator, if a = 1/ mh̄ω (mωx + ip) and a† =
´
³ √
1/ mh̄ω (mωx−ip). show that the harmonic oscillator Hamiltonian
H = (1/2)mω 2 x2 +p2 /2m can be rewritten as H = h̄ω(a† a+1/2), where
[a, a† ] = 1.
2. A laser emits a so-called coherent state of a photon field, defined by
|αi = e−|α|
2 /2
∞
X
αn
n=0
√ |ni,
n!
(1)
where |ni is the normalized state containing n photons. Show that the
coherent state is an eigenstate of the photon annihilation operator, i.
e., show that a|αi = λ|αi, for some constant λ, and find λ.
3. Suppose we have two photon states, described by creation and annihilation operators a† , a, and b† , b. Define a† a − b† b = Z, a† b + b† a = X,
and i(a† b − ab† ) = Y . Show that these operators X, Y , and Z satisfy
the usual commutation relations for Pauli operators.
1
4. Consider a system with two photon modes, as in the previous problem.
Define the Kerr operator K = exp(iχLa† ab† b), where χ and L are
constants. Consider the effect of applying K to a state in which the
a photons are in a coherent state and the b photons are in a state of
definite number. Show that
K|αi|ni = |αeiχLn i|ni,
(2)
where a|αi = α|αi and b† b|ni = n|ni. Thus the Kerr operator just
shifts the phase of the a photons.
5. In discussing NMR quantum computation in class, we considered two
frames of reference: the lab frame, and the rotating frame. If |ψ(t)i is
the time-dependent wave function of a spin-1/2 particle in the lab, then
by definition the wave function |φ(t)i in the rotating frame is related
to |ψ(t)i by
|φ(t)i = exp(iωtZ/2)|ψ(t)i.
(3)
|ψ(t)i satisfies the Schrödinger equation H|ψ(t)i = ih̄(∂/∂t)|ψ(t)i. In
class, we showed that |φ(t)i satisfies the transformed Schrödinger equation H̃|φ(t)i = ih̄(∂/∂t)|φ(t)i, where H̃ = eiωtZ/2 He−iωtZ/2 − (ω/2)Z.
Suppose that H = (h̄ω0 /2)Z + Hac (t), and that ω = ω0 . Find Hac (t)
such that H̃ = g1 (t)X + g2 (t)Y , where g1 (t) and g2 (t) are specified
functions.
2