Bohr–Sommerfeld quantization

Classical Mechanics Level 5

According to the Bohr–Sommerfeld quantization postulate the periodic motion of a particle in a potential field must satisfy the followint quantization rule:

∮ p d q = h n \oint p dq = hn

p p - is momentum of a particle

d q dq - is generalized coordinate of a particle

n n - is a positive integer

Find the value of energy for unidimensional potential field U = k x 2 2 U=\frac{kx^2}{2} at n = 3 n=3 state and

k = 576 π 2 h 2 k=\frac{576\pi^2}{ h^2}

m = 1 m=1


The answer is 36.

Вук Радовић by Вук Радовић , 25, Serbia

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1 solution

By energy conversation:

E = p 2 2 m + U E=\frac{p^2}{2m}+U

Then you will find that p = 2 m E − m k x 2 p=\sqrt{2mE-mkx^2}

So integral is ∫ − a a ⁣ 2 m E − m k x 2 d x = n h \int_{-a}^a \! \sqrt{2mE-mkx^2} dx = nh

where a = 2 E k a=\sqrt{\frac{2E}{k}}

So the final result is

E = n ℏ k m E=n \hbar \sqrt{\frac{k}{m}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Same way and it was a nice problem +1 , semi classical problems really give a preliminary insight into quantum mechanics, also it shows some of the ways in which quantum mechanics was discovered

Mvs Saketh - 6 years, 3 months ago

Good Problem ! I enjoyed in solving this :)

Deepanshu Gupta - 6 years, 3 months ago

I didn't get why the integral take values from -a to a.

Prakhar Gupta - 6 years, 3 months ago

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