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

|z> = e^(-z*z/2) Sum_n (z^n/sqrt[n!]) |n> |

(Had we tried to find eigenstates of a* we would have found they are not normalizable.)

The meaning of z is: z = <z|a|z> = (<x> + i <p>)/sqrt[2] , 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

(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.,

Wed. 10/6:

# Oscillator eigenstates in the position representation:

lowering operator: <x|a|v> = (1/sqrt[2])(x/x_0 + x_0 d/dx)<x|v>

raising operator: <x|a|v> = (1/sqrt[2])(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

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)

where

H_n(y) := exp(+y^2/2) (y - d/dy)^n exp(-y^2/2) defines the

(Note (y-d/dy)exp(+y^2/2) = exp(+y^2/2) d/dy, hence we also have the simpler expression

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).

Fri. 10/8:

# Galilean invariance of the Schrodinger equation, by Prof. Victor Yakovenko:

how does a wavefunction transform when you change to a moving frame, and how can you use this?