Question

Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e. the n-2 wavefunction). (c) By explicit substituion of vix) into Schrodinger's equation with U() Imx2, show that this state has a well-defined energy E?, and determine this energy.

Answer 1

A = 1/ squareroot 2mh w (I P + mwx) = 1/ squareroot 2mhw [h 9/x + mwx] Ay_1 (x) = A/ squareroot 2mhw [h l/x + mwx] xe^-bx^2 = A/ squareroot 2mhw [h(e^-bx^2 -2bx^2e^-bx^2) + mwx^2e^-bx^2] = (a/ squareroot n)^1/2 e^-a^2x^2/2 where a = squareroot mw/h = psi_0 (x) At = 1/ squareroot 2mhw (I p + mwx) = 1/ squareroot 2mhw (-h l/x + mwx) = A/ squareroot 2mhw [-h l/x + mwx] xe^-bx^2 = A/ squareroot 2mhw [-h(e^-bx^2 - 2bx^2 e^-bx^2) + mwx^2 e^-bx^2] = (a/8 squareroot product)^1/2 e^-a^2x^2/2 (4a^2x^2 - 1) where a = squareroot mw/h = psi_2 (x)