Energy Eigenstates. Words by Walter F. Smith, 4-22-08
Tune: “A Comedy
Tonight”, from “A Funny Thing Happened on the Way to the Forum”, by Stephen
Sondheim
1.Take a wavefunction, any wavefunction –
You can synthesize it
out of energy eigenstates!
Why, might you ask,
should I do this task?
Relax, and I will
sing to you of energy eigenstates!
No thinking
hard! Let down your guard!
Once I have sang of
it, you’ll get the hang of it!
Stop feeling queasy –
it’s really easy.
I’ll tell you all
about their traits!
…My favorite
functions, energy eigenstates!
2.To be a bit
formal, they’re orthonormal.
That means the inner product
of two energy eigenstates
Is zero or one, and –
here comes the fun- --
The secret to expand-ing
in energy eigenstates!
All we could
wish! The coeffic-
ient in the sigma is no enigma:
c-n is psi-n star times your psi, then
Integrate it over all
of space!
It’s the inner
product with the energy eigenstate!
3.They’re not
dramatic, in fact they’re quite static.
Nothing ever changes in
an energy eigenstate!
Minus i E-n
t, over h-bar we see
Is the exponential
for an energy eigenstate!
Stately and calm,
they’ve no need for haste.
The lords and ladies
of Hilbert space-!
But add them
up right now, you’ll see a sight now,
With the time dependence
it creates!
Levels electronic!
Spherical harmonics!
Radial equation! Harmonic oscillation!
Tunneling authentic!
Atoms hydrogenic!
Paraboloid, trapezoid, sinusoid, overjoyed!
Stately and calm,
they’ve no need for haste.
The lords and ladies
of Hilbert space-!
But add them
up right now, you’ll see a sight now,
With the time dependence
it creates!
Superpositions of energy eigenstates!