Plank's constant, phase integral and quantization of action

[phonon44145]

I apologize in advance if this is too trivial, but...

Time and again, I hear something along the lines of "Plank's constant is a unit of action", or "Plank's constant is a unit of action in the old quantum theory". In addition, many texts imply some sort of connection between quantization of action and the phase integral. For example, the Eisberg-Resnik textbook (2 ed, p. 112) has this to say about the Wilson-Sommerfeld rule: "The quantity Integral (p dx) is sometimes called a phase integral, in classical physics it is the integral of he dynamical quantity called the action over one oscillation of the motion. Hence, the Plank energy quantization is equivalent to the quantization of action".

Now, as far as I understand Classical Mechanics, the relevant formula for action says

dS = pdq - Hdt

where S is action, p and q are the generalized momentum and coordinate, and H is the Hamiltonian. Then if we require that the phase integral is quantized according to Wilson-Sommerfeld rule, Integral (pdx) = nh, then it only follows that the Plank energy quantization is equivalent to the quantization of the quantity S + Integral (Hdt). How does one derive quantization of S from here?

 

[granpa]

Planck's constant has units of angular momentum

 

[phonon44145]

Obviously. And it also has units of action. But saying "it has units of" is not the same as saying "it is identical to".

 

[Maaneli]

I apologize in advance if this is too trivial, but...

Time and again, I hear something along the lines of "Plank's constant is a unit of action", or "Plank's constant is a unit of action in the old quantum theory". In addition, many texts imply some sort of connection between quantization of action and the phase integral. For example, the Eisberg-Resnik textbook (2 ed, p. 112) has this to say about the Wilson-Sommerfeld rule: "The quantity Integral (p dx) is sometimes called a phase integral, in classical physics it is the integral of he dynamical quantity called the action over one oscillation of the motion. Hence, the Plank energy quantization is equivalent to the quantization of action".

Now, as far as I understand Classical Mechanics, the relevant formula for action says

dS = pdq - Hdt

where S is action, p and q are the generalized momentum and coordinate, and H is the Hamiltonian. Then if we require that the phase integral is quantized according to Wilson-Sommerfeld rule, Integral (pdx) = nh, then it only follows that the Plank energy quantization is equivalent to the quantization of the quantity S + Integral (Hdt). How does one derive quantization of S from here?

You can take dq to be virtual displacements of q, which keeps time fixed.