Chapter 39: Q59P (page 1217)

As Fig. 39-8 suggests, the probability density for the region X>L in the finite potential well of Fig. 39-7 drops off exponentially according to $ψ^{2} (x) = C e^{- 2 k x}$ , where C is a constant. (a) Show that the wave function $ψ (x)$ that may be found from this equation is a solution of Schrödinger’s equation in its one-dimensional form. (b) Find an expression for k for this to be true.

Short Answer

Expert verified

a) It is proved that the wave function $ψ (x)$ is a solution of Schrödinger’s equation in its one-dimensional form.

b) The expression of is $k = \frac{2 π}{h} \sqrt{2 m (U - E)}$ .

Step by step solution

Describe Schrodinger's equation

Schrodinger's equation in the finite potential well at is given by,

$\frac{d^{2} ψ}{{dx}^{2}} + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0 . . . . . . . . (1)$

Show that the wave function ψ(x) is a solution of Schrödinger’s equation in its one-dimensional form(a)

Consider the given wave function.

$ψ^{2} (x) = C e^{- 2 k x} ψ (x) = \sqrt{C} e^{- k x}$

Substitute $ψ (x) = \sqrt{C} e^{- k x}$ in equation (1).

$\frac{d^{2}}{d x^{2}} (\sqrt{C} e^{- k x}) + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0 \sqrt{C} \frac{d}{d x} (- k e^{- k x}) + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0 \sqrt{C} k^{2} e^{- k x} + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0 k^{2} ψ + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0$

Therefore, it is proved that the wave function $ψ (x)$ is a solution of Schrödinger’s equation in its one-dimensional form.

Find an expression for k for this to be true(b)

Consider the following equation.

$k^{2} ψ + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0$

Simplify the above equation for .

$k^{2} ψ = \frac{8 π^{2} m}{h^{2}} (U - E) ψ k^{2} = \frac{8 π^{2} m}{h^{2}} (U - E) k = \frac{2 π}{h} \sqrt{2 m (U - E)}$

Therefore, the expression of is $k = \frac{2 π}{h} \sqrt{2 m (U - E)}$ .

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Short Answer

Step by step solution

Describe Schrodinger's equation

Show that the wave function ψ(x) is a solution of Schrödinger’s equation in its one-dimensional form(a)

Find an expression for k for this to be true(b)

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