Question

In harmonic oscillator transition from ground state to second excited state is forbidden, why?

Answer 1

The transition probability for the transition is zero.
the transition is only possible between states differing by unit change in the n value.

Explanation:

the harmonic oscillator

wave function `psi (x) = C(n) e^-(x^2) . Hn (x)`

where x = y sqrt ( m.w /2h )

, C(n) is a normalization constant, and

Hn(x) are the Hermite Polynomials.

P(n m) , the probability

for a transition between the states n and m, is given by

`P(n m) =` Integral `( psi (x). x . psi(x) .dx )`over the whole space .

naturally the integral will involve

integral of `{ e^-x^2 . x. Hn(x) . Hm(x) dx.}...`

where x goes from -infinity to + infinity.

after calculation it is observed that the integral goes to zero

unless m= n +1 or n-1.

therefore the transition is possible only if m-n = + - 1.