Answered: 7. Write the expressions for momentum…

7. Write the expressions for momentum vector (k), wave function Y (x) and energy of electrons that are free to move between atoms and are experiencing no potential field (V(x) =0).

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Question

7. Write the expressions for momentum vector (k), wave function Y (x) and energy of electrons
that are free to move between atoms and are experiencing no potential field (V(x) =0).

Expert Solution

Step 1

The expression for momentum vector, wave function and energy is to be evaluated.

$\frac{- h^{2}}{8 π^{2} m} \frac{d^{2} ψ}{d x^{2}} + V (x) ψ (x) = E ψ (x) V (x) = 0 \frac{- h^{2}}{8 π^{2} m} \frac{d^{2} ψ}{d x^{2}} + 0 = E ψ (x) \frac{- h^{2}}{8 π^{2} m} \frac{d^{2} ψ}{d x^{2}} = E ψ \frac{d^{2} ψ}{d x^{2}} = \frac{- 8 π^{2} mEψ}{h^{2}} \frac{d^{2} ψ}{d x^{2}} + \frac{8 π^{2} mEψ}{h^{2}} = 0 \frac{d^{2} ψ}{d x^{2}} + k^{2} ψ = 0 (1) k = \sqrt{\frac{8 π^{2} mE}{h^{2}}}$