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phonon44145
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Is it true that in Quantum Mechanics action is quantized?
phonon44145 said:Is it true that in Quantum Mechanics action is quantized?
phonon44145 said:I think it would be instructive to consider the 1-D oscillator first. As they show in most textbooks, we would draw an ellipse in the phase space, p^2/2mE + x^2/(2E/k) = 1, and then Wilson-Sommerfeld quantization Int pdx = nh will immediately lead to Plank's quantization law E=nhf. But why do they say that Int pdx is equivalent to action over one oscillation? Action in Classical Mechanics is dS=pdq - Hdt, so if we quantize Int pdx this means we have quantized the quantity S + Int Hdt, but we have not quantized S. So where does action quantization come from?
Ok, thanks. So energy.time is not conventionally quantized or quantizable. Is that correct?atyy said:Planck's constant has units of action.
In Bohr-Sommerfeld quantization, the action is quantized in multiples of Planck's constant. However, that should be seen as part of "old quantum physics" like the Bohr atom. It's great for intuition, but after Heisenberg, Schroedinger, Dirac, quantization is specified by making canonically conjugate variables not commute. The old quantum physics is an approximation to the results of the proper quantum formalism.
Thanks for the feedback.phonon44145 said:ThomasT,
I don't think it's that simple. True, h has units of action (J*s), but that fact by itself does not mean that h IS action. Now, action itself is a classical concept. For example, in classical mechanics it is S=Int L dt where L is Lagrangian. In quantum mechanics, the duration of a process is indeterminate due to energy-time uncertainty principle. So if we know L exactly, then how can we know the limits of integration? This issue did not exist in the old (pre-Heisenberg) quantum theory. But even in the old theory, Wilson-Sommerfeld only assumed quantization of the phase integral Int p dq which does not imply quantization of S (unless you make an additional assumption that Int H dt is also quantized). To complicate matters further, Plank spoke of quantized action well before Sommerfeld, i.e. without access to the formula Int p dq = nh. What was his reasoning then? Was it just the fact that h and S had the same units?
ThomasT said:Ok, thanks. So energy.time is not conventionally quantized or quantizable. Is that correct?
Thanks, I will read Demystifier's section on this. Meanwhile, I remain fond of useful heuristics.atyy said:Yes, in the strict sense that there is no time operator, and no commutation relation between energy and time. This is explained in Demystifier's http://arxiv.org/abs/quant-ph/0609163 (see his section 3).
However, it is useful to have an energy-time uncertainty relation as a heuristic.
ThomasT said:Thanks, I will read Demystifier's section on this. Meanwhile, I remain fond of useful heuristics.
Quantization in physics refers to the process of representing continuous physical quantities in discrete or discrete values. In the case of action, it means that the amount of action that can be observed in a physical system is limited to specific, discrete values rather than being continuous.
In quantum mechanics, action is quantized due to the wave-particle duality of matter. According to the uncertainty principle, it is not possible to know the exact position and momentum of a particle simultaneously. This uncertainty leads to the quantization of action, as the particle's position and momentum can only take on certain discrete values.
No, the quantization of action is only observed in systems that operate on a quantum scale, such as atoms and subatomic particles. Macroscopic systems, such as everyday objects, do not exhibit quantized action.
In atoms, the quantization of action leads to the existence of discrete energy levels. As electrons move between these energy levels, they emit or absorb photons with specific energies, resulting in the emission or absorption of light at specific wavelengths. This phenomenon is known as spectral lines and is a direct result of the quantization of action.
The quantization of action is one of the fundamental principles of quantum mechanics and has revolutionized our understanding of the physical world. It has led to discoveries such as the wave-particle duality of matter, the uncertainty principle, and the existence of discrete energy levels. Without the quantization of action, our current understanding of the behavior of matter and energy would not be possible.