All Problems

Wave Properties of particles. Schrodinger Equation.

Problem 6.90

The wave function of a particle of mass mm in a unidimensional potential field U(x)=kx2/2U(x)=k x^{2} / 2 has in the ground state the form ψ(x)=Aeαx2,\psi(x)=A \mathrm{e}^{-\alpha x^{2}}, where AA is a normalization factor and α\alpha is a positive constant. Making use of the Schrödinger equation, find the constant α\alpha and the energy EE of the particle in this state.

Reveal Answer
α=mω/2,E=ω/2, where ω=k/m\alpha=m \omega / 2 \hbar, E=\hbar \omega / 2, \text { where } \omega=\sqrt{k / m}