Bohr-Sommerfield quantization of motion

[weezy]

From wikipedia I understand that the old quantum condition $$\oint_{H(p,q)=E} p_i dq_i = n_i h $$ states that not all kinds of motion are permitted in a system. My question is why is this called the old quantum condition and what is quantization of motion? Does this mean that a particle jumps from velocity $v_1 \rightarrow v_2$ in a certain step ? I am mostly familiar with Schrödinger's equation which can be solved to see that Energy levels of a particle in a box are quantized. But does energy quantization imply momentum quantization as well? I understand the motivation of this principle comes from visualizing a particle as a wave but most experienced physicists say that is an outdated concept. If that is the case then what's the newer approach or the new quantum condition?

 

[stevendaryl]

From wikipedia I understand that the old quantum condition $$\oint_{H(p,q)=E} p_i dq_i = n_i h $$ states that not all kinds of motion are permitted in a system. My question is why is this called the old quantum condition and what is quantization of motion? Does this mean that a particle jumps from velocity $v_1 \rightarrow v_2$ in a certain step ? I am mostly familiar with Schrödinger's equation which can be solved to see that Energy levels of a particle in a box are quantized. But does energy quantization imply momentum quantization as well? I understand the motivation of this principle comes from visualizing a particle as a wave but most experienced physicists say that is an outdated concept. If that is the case then what's the newer approach or the new quantum condition?

The original idea was that an electron orbited the nucleus of an atom in the same way it would in classical mechanics, except that there was an additional restricting the orbits to those satisfying the quantization condition. The hypothesis was that if you tried to add energy to the system, the system would not absorb the energy in discrete quantities such that the quantization condition held before and after the absorption. (This was the "quantum leap" from one energy level to another).

The new quantum condition is given by Schrödinger's equation--the energy can only take on those values for which the time-independent Schrödinger equation can be solved to produce a square-integrable function.

 

[weezy]

The new quantum condition is given by Schrödinger's equation--the energy can only take on those values for which the time-independent Schrödinger equation can be solved to produce a square-integrable function.

You mean the stationary states?

 

[A. Neumaier]

For the old quantum theory see https://en.wikipedia.org/wiki/Old_quantum_theory and https://www.physicsforums.com/posts/5588027
Its modern form is called semiclassical quantum mechanics.