Question

7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the matrix elements (nlpIn') 3. In a basis where the In) form basis vectors, we can construct an infinite-dimensional matrix representation of operators using the matrix elements. Construct the infinite- dimensional matrix x in this basis. 4. Construct the infinite-dimensional matrix p in this basis 5. Show that the infinite-dimensional matrix H (1/2m)p(mu/2)x2 is diagonal in this basis. What are the diagonal elements?

Answer 1

x^^= squareroot h/2 mw (a^+ + a^.) < n Vertical-bar x^^ Vertical-bar n^. > = squareroot h/2 mw < n Vertical-bar a^+ + a Vertical-bar n' > = squareroot h/2 mw [ < n Vertical-bar a^+ Vertical-bar a' > < n Vertical-bar a Vertical-bar n' > ] = squareroot h/2 mw [squareroot h' + 1) < n Vertical-bar n' + 1 > + squareroot n' < n Vertical-bar n' - 1 > ] = squareroot h/Partialdifferential mw [squareroot n' + 1 delta_n1 n' + 1 + squareroot n' delta_n1 n' - 1] Therefore delta_m, n = {1 if m = n 0 if m notequalto n x_n' = ( < 0 Vertical-bar x Vertical-bar 0 > < 1 Vertical-bar n Vertical-bar 0 > < 2 Vertical-bar x Vertical-bar 0 > < 0 Vertical-bar x Vertical-bar 1 > < 1 Vertical-bar x Vertical-bar 1 > < 2 Vertical-bar x Vertical-bar 1 > < 0 Vertical-bar x Vertical-bar 2 > < 1 Vertical-bar x Vertical-bar 0 > < 2 Vertical-bar n Vertical-bar 2 > < 0 Vertical-bar x 2 Vertical-bar n > < 1 Vertical-bar x^2 Vertical-bar n > < 2 Vertical-bar x Vertical-bar n > ) \r\n x_mn' = (0 pi 0 squareroot 1 0 squareroot 2 0 squareroot 2 0 0 0 squareroot 3 0 0 0 0) squareroot h/2nw p^^- i squareroot h mw/Partialdifferential (a^+ - a) < n Vertical-bar p Vertical-bar n') = i squareroot h/Partialdifferential mw [ < n Vertical-bar a^+