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Wave Properties of particles. Schrodinger Equation.

Problem 6.73

A particle of mass mm moves in a unidimensional potential field U=kx2/2U=k x^{2} / 2 (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field.

Reveal Answer
6.73. Taking into account that p∼Δp∼ℏ/Δx∼ℏ/x,p \sim \Delta p \sim \hbar / \Delta x \sim \hbar / x, we get E=T+U≈ℏ2/2mx2+kx2/2.E=T+U \approx \hbar^{2} / 2 m x^{2}+k x^{2} / 2 . From the condition dE/dx=0d E / d x=0 we find x0x_{0} and then Emin⁡≈ℏk/m=ℏω,E_{\min } \approx \hbar \sqrt{k / m}=\hbar \omega, where ω\omega is the oscillator's angular frequency. The rigorous calculations furnish the value ℏω/2\hbar \omega / 2
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