sci.physics.research

On 2007-08-04, Jay R. Yablon wrote:
>
> "Igor Khavkine" < @ > wrote in message
> news: @ ...
>> On Jul 31, 3:33 pm, "Jay R. Yablon" wrote:
>>> Insofar as I understand, the action S = $d^4x L, where L is the
>>> Lagrangian density, is always a multiple of 1/2 unit of the reduced
>>> Planck's constant ($ denotes an integral), that is:
>>>
>>> S = $d^4x L = +/- .5 n h-bar (1)
>>>
>>> Is this an accurate statement?
>>
>> No, if you intend n to represent an integer.
>>
> Yes, n is an integer. So, why not [?]

Same reason that 1 is not equal to 2. It is simply not true.

> and what IS the correct way to
> express quantization of the action?
> [...]
> Or, something else?

Who says that the value of the action functional is quantized in the
first place?

Since you haven't provided a reference to such an assertion, I'd have to
guess as to its origin. It is most likely that you've confused two
different uses of the word "action". In one case, we have an "action
functional", whose expression you've given above. In another case, we
have an "action variable", which usually comes together with an "angle
variable". The definition of action angle variables can be found in many
texts on mechanics. See for instance sections 36 and 37 of [1] or
sections in [2]. Both "actions" happen to have the same units,
though. The same units as angular momentum.

The old quantum theory, called the Sommerfeld-Wilson quantization rule,
prescribes that for a quantum system of a single degree of freedom,
undergoing periodic motion, the action variable J of the system may have
only discrete values given by

J = h n,

where h is the Planck constant and n is a non-negative integer. This
"rule" is known not to be precisely correct and has been supplanted by
the modern quantum mechanical formalism (Hilbert spaces, operators,
canonical quantization, and so on). This rule still survives as an
asymptotic expansion for the spectrum of the corresponding canonically
quantized operator J. More precisely, this "rule" states that the
eigenvalues of J will be discrete and, for large n, will have the form
J_n ~ h (n+w) + O(h^2), where w is a corrective constant originally
ignored by Sommerfeld and Wilson. The proof of this asymptotic formula
is given through the WKB approximation, wich also allows one to compute
the O(h^2) and higher order expansion terms.

For the simple harmonic oscillator, w = 1/2, the O(h^2) terms happen to
identically vanish and the action variable J is proportional to the
total energy, yielding the well known energy spectrum E = h-bar omega
(n+1/2). Any text which explains the WKB approximation gives many more
examples.

I hope this explanation underscores the importance of context when
trying to interpret or use any particular "law" or "rule" in physics.

[1] Fetter & Walecka, Theoretical Mechanics of Particles and Continua,
Dover (2003).
[2] Goldstein, Classical Mechanics (2nd ed.), Addison-Wesley (1980).


Igor