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Wave Properties of particles. Schrodinger Equation.

Problem 6.87

Find the possible values of energy of a particle of mass mm located in a spherically symmetrical potential well U(r)=0U(r)=0 for r<r0r< r_{0} and U(r)=∞U(r)=\infty for r=r0,r=r_{0}, in the case when the motion of the particle is described by a wave function ψ(r)\psi(r) de pending only on rr Instruction. When solving the Schrödinger equation, make the substitution ψ(r)=χ(r)/r\psi(r)=\chi(r) / r

Reveal Answer
6.87. Utilizing the substitution indicated, we get χ′′+k2χ=0, where k2=2mE/ℏ2 \chi^{\prime \prime}+k^{2} \chi=0, \quad \text { where } \quad k^{2}=2 m E / \hbar^{2} We shall seek the solution of this equation in the form χ=\chi= =asin⁡(kr+α)=a \sin (k r+\alpha). From the finiteness of the wave function ψ\psi at the point r=0r=0 it follows that α=0.\alpha=0 . Thus, ψ=(a/r)sin⁡kr.\psi=(a / r) \sin k r . From the boundary condition ψ(r0)=0\psi\left(r_{0}\right)=0 we obtain kr0=nπ,k r_{0}=n \pi, where n=1,2,… Hence, En=n2π2ℏ2/2mr02 n=1,2, \ldots \text { Hence, } E_{n}=n^{2} \pi^{2} \hbar^{2} / 2 m r_{0}^{2}
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