Question

Potential energy function,

V(x) = (1/2)mw²x²

Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the 6

Answer 1

Time independent Schrodinger equation for one dimensional harmonic oscillator is given as -n^2/2m d^2/dx^2 + 1/2 m omega^2 x^2 psi = psi rightarrow (1) given psi_0 = co exp [-x^2/2x^2] rightarrow d psi_0 /dx = -x/x^2 co exp [-x^2/2x^2] rightarrow d^2/d x^2 = -co/x^2 exp[-x^2/2x^2] + x^2/alpha^4 co exp [-x^2/2x^2] = -psi/alpha^2 + x^2/alpha^4 psi _0 rightarrow 3 substituting (2) and (3) i8nto (1) -n^2/2m [-psi_0/alpha^2 + x^2/alpha^4 psi_0] + 1/2 m omega ^2 [psi_0] = E_0 psi_0 rightarrow n^2/2n alpha^2 - n^2 x^2/2m alpha^4 + 1/2 m omega^2 x^2 = E_0 rightarrow (4) given alpha^2 = n/m times (m omega^2)^1/2 = h/m omega also , E_0 = 1/2 h omega eq (4) therefore , can be written as: \r\n n^2/2m (n/m omega) + n^2/m(n/m omega) - x^2 n^2/2m (h/m omega)^2 + 1/2 m omega^2 + 1/2 m omega^2 x^2 = 3/2 n omega rightarrow 1/2 n omega + h omega - x^2 m omega^2/2 + 1/2 m omega^2 x^2 = 3/2 n omega rightarrow 3/2 n omega = 3/2 n omega