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Wave Properties of particles. Schrodinger Equation.

Problem 6.76

Find a particular solution of the time-dependent Schrödinger equation for a freely moving particle of mass mm.

Reveal Answer
6.76.6.76 . The solution of the Schrödinger equation should be sought in the form Ψ=ψ(x)⋅f(t).\Psi=\psi(x) \cdot f(t) . The substitution of this function into the initial equation with subsequent separation of the variables xx and tt results in two equations. Their solutions are ψ(x)∼eikx\psi(x) \sim \mathrm{e}^{i k x}, where k=2mE/ℏ,Ek=\sqrt{2 m E} / \hbar, E, is the energy of the particle, and f(t)∼f(t) \sim ∼e−iωt,\sim \mathrm{e}^{-i \omega t}, where ω=E/ℏ.\omega=E / \hbar . Finally, Ψ=aei(kx−ωt),\Psi=a \mathrm{e}^{i(k x-\omega t)}, where aa is a certain constant.
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