Class calendar, week 6 Mon. 10/4:
# In oscillator ground state, <0|a|0>=0, hence <x>=0=<p>.
The ground state is a minimum uncertainty wavepacket:
the condition a|0>=0 implies the minimum uncertainty condition, p'|0>=i
r x'|0>, with r = hbar/x_0^2.
As seen above (9/27) therefore <x|0> is a Gaussian, exp(-x^2/2x_0^2).
# Are there any other eigenstates of a, i.e. a|z> = z|z>?
a is not hermitian, so there need not be more eigenstates, and,
if there are, the eigenvalue z need not be real. Solving we find the answer
is yes. These are called coherent states:
|z> = e^(-z*z/2) Sum_n (z^n/sqrt[n!])
As defined here <z|z>=1.
(Had we tried to find eigenstates of a* we would have found they are
The meaning of z is: z = <z|a|z> = (<x> + i <p>)/sqrt ,
in units with x_0=1=hbar.
|z> is also a minimum uncertainty state, with the same width of the
Gaussian as the ground state,
but translated in x and p by <x> and <p>.
# Coherent states are never orthogonal to each other, however they
do span the Hilbert space.
Explicitly, <a|b> = exp(a*b - a*a/2 - b*b/2), and \int (d^2z/pi)
|z><z| = I.
# Evolution: The number eigenstates are energy eigenstates and evolve
trivially: |n,t> = e^(-i(n+1/2)wt) |n>.
Putting this in the defining sum we find that the coherent states evolve
as |z,t> = e^(-iwt/2) |e^(-iwt) z>,
i.e., they evolve to new coherent states.
# This conclusion is also easily reached in the Heisenberg picture,
where we find a(t) = e^(-iwt) a(0).
# The evolution of <x> and <p> implied by z(t) = e^(-iwt) z(0)
must be just the classical evolution, since
the Heisenberg equation of motion for a is identical to the classical
equation of motion,
and a is linear in x and p.
# It is neat that coherent states evolve to coherent states, but is
it useful? YES! The reason is that
they remain coherent states even in the presence of an arbitrary
time-dependent homogeneous force. (You show this in hw5.)
In particular, if you start with the ground state and drive it with
such a force, you get a coherent state.
For example, a mode of the electromagnetic field, initially in the
vacuum state, will become a coherent
state when driven by an external current. Another nice example is in
the paper I will hand out, where
experimenters trapped and cooled a Beryllium ion to the ground
state of the trap potential, and then excited
the translational motion by laser pulses. See C.
Monroe et. al., A "Schrodinger Cat" Superposition State of an Atom,
Science 272 (24 May 1996) 1131, or Quantum
harmonic oscillator state synthesis and analysis, quant-ph/9702038.
# Oscillator eigenstates in the position representation:
lowering operator: <x|a|v> = (1/sqrt)(x/x_0 + x_0 d/dx)<x|v>
raising operator: <x|a|v> = (1/sqrt)(x/x_0
- x_0 d/dx)<x|v>
Ground state: <x|a|v> = 0 => (y + d/dy)<y|v>=0 where we define
the dimensionless position y=x/x_0.
Normalized solution is <x|v> = (sqrt[pi] x_0)^-1/2 exp(-y^2/2).
Excited states: <x|n> = (n!)^-1/2 <x|(a*)^n|0>
= (sqrt[pi] x_0 2^n n!)^-1/2 (y - d/dy)^n exp(-y^2/2)
= (sqrt[pi] x_0 2^n n!)^-1/2 H_n(y) exp(-y^2/2)
H_n(y) := exp(+y^2/2) (y - d/dy)^n exp(-y^2/2) defines the Hermite
(Note (y-d/dy)exp(+y^2/2) = exp(+y^2/2) d/dy, hence we also have the
H_n(y) = exp(+y^2/2) (y - d/dy)^n exp(+y^2/2)exp(-y^2)
= exp(+y^2) (-d/dy)^n exp(-y^2).)
Orthonormality of the number eigenstates <n|m>=delta(n,m) implies
for the Hermite polynomials
\int dy e^(-y^2) H_n(y) H_m(y) = (sqrt[pi] x_0 2^n n!) delta(n,m).
# H_0(y) = 1, H_1(y) = 2y, H_2(y) = 4y^2 -2,
H_3(y) = 8y^3 - 12y, H_4(y) = 16y^4 - 48 y^2 +12.
# H_n(y) is a polynomial of order n, so the parity of H_n is (-1)^n
and H_n(y) = 0 has n roots.
In fact the roots are all real and distinct, but I don't see a simple
way to see this directly from the definition.
# Showed CUPS computer simulations of the time evolution of Gaussian
wavepackets for free particles
and oscillators, and coherent state evolution for an oscillator (including
the driven case).
# Galilean invariance of the Schrodinger equation, by Prof. Victor
how does a wavefunction transform when you change to a moving frame,
and how can you use this?