Without, actually doing any integrals determine whether the following integrals

Question

Without, actually doing any integrals determine whether the following integrals involving harmonic: oscillator wave functions are equal to 0, not zero, or cannot be determined. If they can t be determined, explain what additional information you would need to have to make the determination. Integral_-infinity^infinity psi_1^*psi_2dx Integral_-infinity^infinity psi_1^*x^cappsi_1dx Integral_-infinity^infinity psi_1^*x^cap2psi_1dx Integral_-infinity^infinity psi_1^*psi_3dx Integral_-infinity^infinity psi_1^*V^cap psi_1dx

Explanation / Answer

a) For harmonic oscillators, the two states are orthogonal to each other so their inner product is zero.

The given integral gives the inner product between ground and first excited states which are different. So it is equal to zero.

b) The integral gives the average value of position. Here the wave function is even and bu multipling with x it become odd. The integral is from -infinity to infinity. So the integral is equal to zero.

<x> = 0

c) The integral gives the average value of x^2 so that the net function is even so this integral is not zero.

d) This integral gives the inner product of first and third states whch are orthogonal. So the integral is zero.

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