Example 8B.2 Normalizing a harmonic oscillator wavefunction
Find the normalization constant for the harmonic oscillator wavefunctions.
Method Normalization is carried out by evaluating the integral of |Ή|² over all space and then finding the normalization factor from eqn 7B.3 (N=1/(∫Ή*Ήdτ)¹/²). The normalized wavefunction is then equal to NΉ. In this one-dimensional problem, the volume element is dx and the integration is from -∞ to +∞. The wavefunctions are expressed in terms of the dimensionless variable y=x/α, so begin by expressing the integral in terms of y by using dx=αdy. The integrals required are given in Table 8B.1.
Answer The unnormalized wavefunction is
ψv(x)=Hv(y)e^-y²/2
It follows from the integrals given in Table 8B.1 that
∫₋∞⁺∞ψv*ψvdx=α∫₋∞⁺∞ψv*ψvdy=α∫₋∞⁺∞ Hv²(y)e^-y²dy=απ¹/²2ᵛv!
where v!=v(v-1)(v-2)...1. Therefore,
Nv = (1 / (απ¹/²2ᵛv!))¹/²
Note that, unlike the normalization constant for a particle in a box, for a harmonic oscillator Nv is different for each value of v.
Confirm, by explicit evaluation of the integral, that ψ0 and ψ1 are orthogonal.
Answer: Show that ∫₋∞⁺∞ ψ0*ψ1dx=0 by using the information in Table 8B.1.