Harmonic Oscillators: Raising and lowering operators

Harmonic oscillator Hamiltonian in one dimension. Ground state (Subscript[ψ, 0]). Check normalization. Plot ground state, evaluated at ...

Hints: The shortcut for subscripts is “Ctrl -“, and superscripts is “Ctrl ^”. The shortcut for ℏ is “esc hb esc”, ∞ is “esc inf esc”, π is “esc p esc”, and ω is “esc w esc”.

Define (Subscript[ψ, 0]) symbolically. Is it normalized?

(Subscript[ψ, 0]) := (m ω/(π ℏ))...;
Integrate[...,{x,-∞,∞}, Assumptions->{m>0,ω>0,ℏ>0}]

We take our constants from McEuen’s bouncing buckyballs, Park et al., “Nanomechanical oscillations in a single-C60 transistor”, Nature 407, 57 (2000).

amu = 1.660538782×10^−24;
constants = {m -> ..., ℏ->1.054571628251774 10^-27, ...};
a0= .../.constants;(* RMS width of Gaussian *)
(Subscript[ψ, 0]) /. constants // Simplify

Plot[(Subscript[ψ, 0])/.constants, {x, ..., ...}, PlotRange->All]

How do the zero-point fluctuations for McEuen' s bouncing buckyball compare to the size of an atom?

Note the Hamiltonian is an operator mapping functions into functions. H[ψ] is a new function. We define a function on function space that returns H[ψ]. You can take the second derivative of ψ as D[ψ, {x, 2}].

H[ψ_]:= ...

H[(Subscript[ψ, 0])]/(Subscript[ψ, 0]) // Simplify

Momentum operator p. Creation operator (SuperPlus[a]). Exciting the ground state using creation operators.

p[ψ_] :=...
(SuperPlus[a])[ψ_] := ...

H[(SuperPlus[a])[(Subscript[ψ, 0])]]/...// Simplify

Integrate[...]

(SuperPlus[a])[(SuperPlus[a])[(Subscript[ψ, 0])]]//Simplify

Integrate[...]

This isn’t the eigenstate (Subscript[ψ, 2]), since it has the wrong norm. Try taking the norm of higher powers of (SuperPlus[a])[(SuperPlus[a])...[(Subscript[ψ, 0])]]...
 until you figure out what to divide by to normalize it.

Integrate[...]

Defining normalized eigenstates. Plots.

Hint: We are using recursion, defining (Subscript[ψ, n]) in terms of (Subscript[ψ, n-1]). It should be (SuperPlus[a])(Subscript[ψ, n-1]) up to a constant that depends on n. You can either figure this out analytically using commutation relations, or by working by analogy from the norms above...

(Subscript[ψ, n_]) :=...(SuperPlus[a])[(Subscript[ψ, n-1])]//Simplify

Integrate[(Subscript[ψ, 4])^2, ...]

{(Subscript[ψ, 0]),(Subscript[ψ, 1]),(Subscript[ψ, 2]),(Subscript[ψ, 3]),(Subscript[ψ, 4])}

Wavefunctions = {(Subscript[ψ, 0])/.constants,(Subscript[ψ, 1])/.constants,(Subscript[ψ, 2])/.constants,(Subscript[ψ, 3])/.constants}

Plot[Wavefunctions, {x, ...}, PlotRange->All]