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*From*: John Denker <jsd@av8n.com>*Date*: Fri, 2 Nov 2018 08:13:14 -0700

On 11/2/18 7:39 AM, Carl Mungan wrote:

In intro quantum mechanics, we explain that the translational

(particle in a box) or vibrational (simple harmonic oscillator)

ground-state energy cannot be zero because of Heisenberg’s

Uncertainty Principle (HUP). This is also reinforced by choosing the

ground-state quantum number to be 1 not 0.

We, Kemosabe?

The uncertainty principle doesn't actually say any of those

things.

As a separate matter, FWIW, the ground state quantum number

for a harmonic oscillator is conventionally N=0, not N=1.

The energy is proportional to N+½.

However, the same argument doesn’t hold for the rotational (rigid

rotor) ground-state energy which is zero and which is labeled with

quantum number 0.

An alert student might ask: Why doesn’t HUP apply? If the energy were

zero, wouldn’t I simultaneously know the exact angular position and

angular speed?

The angular speed is zero, but you don't know anything

whatsoever about the angle. There is no violation of

the uncertainty principle.

When I think further about what the ground-state wavefunctions

actually look like, it seems to me that a key difference is that in

the translational or vibrational case, the wavefunction has to go to

zero at the two ends (either the two walls for an infinite wall, or

at +/-infinity for a finite well or oscillator), whereas we impose

periodic boundary conditions (BCs) instead for the rotational case.

That's the right way to approach it.

The wavefunction has to be nonzero somewhere.

It has to go to zero at the wall (for a particle in a box)

or eventually (for a harmonic oscillator).

Therefore it has some curvature.

Therefore it has some nonzero kinetic energy.

This proves that the ground-state energy is nonzero,

without mentioning the uncertainty principle.

But how does this difference in BCs factor into HUP?

It would be simpler to ask how the BCs factor into the assertion

of nonzero energy; this question is answered above.

As for using such scenarios to shed light on the uncertainty

principle, I'm not sure that's possible. It might be, but I'd

have to think about it some more. Hmmmmm.

**Follow-Ups**:**Re: [Phys-L] ground-state energies***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-L] ground-state energies***From:*Carl Mungan <mungan@usna.edu>

**References**:**[Phys-L] ground-state energies***From:*Carl Mungan <mungan@usna.edu>

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