"Exercise 1. Verify eigenstates and eigenvalues." Hhat(f) = phat(phat(f)) / (2 m) + V f phat(f) = -i hbar d(f,x) V = m omega^2 x^2 / 2 psi(n) = C(n) exp(-m omega x^2 / (2 hbar)) H(n, x sqrt(m omega / hbar)) C(n) = 1 / sqrt(2^n n!) (m omega / (pi hbar))^(1/4) H(n,y,z) = (-1)^n exp(y^2) eval(d(exp(-z^2),z,n),z,y) E(n) = hbar omega (n + 1/2) check(Hhat(psi(0)) == E(0) psi(0)) check(Hhat(psi(1)) == E(1) psi(1)) check(Hhat(psi(2)) == E(2) psi(2)) check(Hhat(psi(3)) == E(3) psi(3)) check(Hhat(psi(4)) == E(4) psi(4)) "ok"
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