Quantum Mechanics | Part IB, 2002

Consider a quantum mechanical particle of mass mm moving in one dimension, in a potential well

V(x)={,x<00,0<x<aV0,x>aV(x)=\left\{\begin{array}{cr} \infty, & x<0 \\ 0, & 0<x<a \\ V_{0}, & x>a \end{array}\right.

Sketch the ground state energy eigenfunction χ(x)\chi(x) and show that its energy is E=2k22mE=\frac{\hbar^{2} k^{2}}{2 m}, where kk satisfies

tanka=k2mV02k2.\tan k a=-\frac{k}{\sqrt{\frac{2 m V_{0}}{\hbar^{2}}-k^{2}}} .

[Hint: You may assume that χ(0)=0.]\chi(0)=0 .]

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