Question

(a) Show that the wave-function ψ₀(x) = A exp(−x²/2a²) with energy E = ω/2 (where A and a are constants) is a solution for all values of x to the one-dimensional time-independent Schrodinger equation (TISE) for the simple harmonic oscillator (SHO) potential V(x) = mω²x²/2 

(b) Sketch the function ψ1(x) = Bx exp(−x²/2a²) 

(where B = constant), and show that it too is a solution of the TISE for all values of x. 

(c) Show that the corresponding energy E = (3/2)ℏω 

(d) Determine the expectation value < p_x > of the momentum in state ψ₁ 

(e) Briefly discuss the relevance of the SHO in describing the behavior of diatomic molecules. 

Answer 1

(a) ψ₀(x) = A exp(−x^2/2a²) Differentiate twice and multiply by −ℏ²/2m

= zero (because integration over an odd function between symmetrical limits is zero). This result is expected because half of the time the particle will be pointing along positive direction and for the half of time in the negative direction. 

(e) The results of SHO are valuable for the analysis of vibrational spectra of diatomic molecules, identification of unknown molecules, estimation of force constants etc.