9:09 a.m.
Friday 11 March, still 1999+17, but who can really believe it?
J.S. Townsend discusses the absence of
oscillatory behavior in the quantum harmonic oscillator, p.261. Transitions
between energy states in the quantum oscillator produce radiation—sorry, not
just the quantum oscillator, but the electrically charged quantum
oscillator. That’s why it’s so strange
to read these two sentences in Townsend’s book:
“A harmonic oscillator in an energy eigenstate is in a stationary state.
Thus it will not exhibit the characteristic oscillatory behavior of a classical
oscillator.”
The quantum
harmonic oscillator is the behavior Planck discovered, the behavior on which
Bohr’s H atom model depends, the very basic QUANTUM behavior itself. So to read it stated so bluntly is
strange: the quantum harmonic oscillator
does not oscillate!
Townsend is
in my opinion a pedagogical genius for simply stating this idea. Also, he writes very well and his book is
organized in a way I like. After the above two statements, he says, “Time
dependence for the harmonic oscillator results from the system being in a superposition of energy eigenstates with
different energies.” His emphasis, not
mine. But, of course, in Q.M. the emphasis is always on the strangeness of its
requirement that coherent superpositions exist. Why is this a requirement in
Q.M.? (My emphasis this time.)
This absence of time dependence relates to the elementary quantum mechanics class at UT-Austin I took in 1998 taught by Weinberg (yes, "the" Weinberg). I was an older-than-average student, and asked more questions than the others, who asked very few themselves. Weinberg was in a seminar lecturing mode that didn't encourage questions from the class. One of my questions was about the quantum harmonic oscillator. After Weinberg had written the equation for the allowed energy states, and drawn the standard energy level picture on the board, I asked something about the stability of the states. He said that they were "absolutely stable." I asked if the quantum oscillator could "emit energy," and Weinberg said yes, "if you give it a way of emitting energy." These quotes are from my class notes of September 15, 1998. I didn't ask for an example of giving the quantum oscillator a way of emitting energy. Time-dependent perturbation theory is generally the way to do it, or to calculate any time-dependent quantity that can be considered to be the result of a small disturbance to the otherwise stable system.
From my notes made after class on September 3rd, I found this comment of mine: "How was The Weinberg? Accessible, articulate in a simple-language kind of way, and thoughtful. Not pedantic. So far (3 days worth), so good."
I plan to devote a lot of writing in this blog to how you can give an electrically charged harmonic oscillator a way of emitting electromagnetic energy. I'll start by finishing up the Planck normal modes electromagnetic cavity radiation discussion sometime soon. This is not a perturbation problem, it's an equilibrium problem, and as discussed in the above journal entry of mine, involves a superposition of states.
----------------------------------end of journal entry-------------------------------------------
From my notes made after class on September 3rd, I found this comment of mine: "How was The Weinberg? Accessible, articulate in a simple-language kind of way, and thoughtful. Not pedantic. So far (3 days worth), so good."
I plan to devote a lot of writing in this blog to how you can give an electrically charged harmonic oscillator a way of emitting electromagnetic energy. I'll start by finishing up the Planck normal modes electromagnetic cavity radiation discussion sometime soon. This is not a perturbation problem, it's an equilibrium problem, and as discussed in the above journal entry of mine, involves a superposition of states.
It's interesting to think about the difference in the time dependence of the oscillator resulting from a superposition of different energy eigenstates and the time dependence of a system like a Schrödinger's cat state, which results from the system's inherent time dependence--the unstable nature of the radioactive atom in the box--rather than the imposed superposition time dependence of the oscillator.
Here's an update on 22 July 1998+18: I've been reading in Griffiths Intro to QM book again recently, and the very first and very simplest example of time dependence is discussed in his Example 2.1, page 29, 2nd ed. Using an initial, t=0, wave function ɸ(x,0) = a superposition of two states, and multiplying this by the usual time-dependent complex exponential function exp[-itEn/hbar] gives a probability density with cos[(E2 – E1)t/hbar] time dependence. If we allow (E2 – E1) = hbarω, this is a quantum harmonic oscillator.
Griffiths says, "notice it took a linear combination of states (with different energies) to produce motion." Well, yeh, and look what kind of motion! But is it emitting energy? No. It's just a stable, single-frequency oscillator. What is oscillating? The probability density. Can you call that motion? Of what? "The system." Later!
Here's an update on 22 July 1998+18: I've been reading in Griffiths Intro to QM book again recently, and the very first and very simplest example of time dependence is discussed in his Example 2.1, page 29, 2nd ed. Using an initial, t=0, wave function ɸ(x,0) = a superposition of two states, and multiplying this by the usual time-dependent complex exponential function exp[-itEn/hbar] gives a probability density with cos[(E2 – E1)t/hbar] time dependence. If we allow (E2 – E1) = hbarω, this is a quantum harmonic oscillator.
Griffiths says, "notice it took a linear combination of states (with different energies) to produce motion." Well, yeh, and look what kind of motion! But is it emitting energy? No. It's just a stable, single-frequency oscillator. What is oscillating? The probability density. Can you call that motion? Of what? "The system." Later!
