Chapter 5: Q92CE (page 193)

a) Taking the particle’s total energy to be 0, find the potential energy.
(b) On the same axes, sketch the wave function and the potential energy.
(c) To what region would the particle be restricted classically?

Short Answer

Expert verified

The potential energy is $U = \frac{h^{2}}{m} \frac{3 x^{2} - a^{2}}{m {(x^{2} + a^{2})}^{2}}$

Step by step solution

Potential energy

(a) If the total energy is 0, the potential energy will be the negative kinetic energy; the kinetic energyhas an operator.

_{$\hat{E} = - \frac{h^{2}}{2 m} \frac{d^{2}}{{dx}^{2}}$}

The kinetic energy is

_{$\hat{E} (\sqrt{\frac{2}{Π}} a^{3 / 2} \frac{1}{x^{2} + a^{2}}) = - \frac{h^{2}}{2 m} \frac{d^{2}}{{dx}^{2}} (\sqrt{\frac{2}{Π}} a^{3 / 2} \frac{1}{x^{2} + a^{2}})$}

_{$= \frac{h^{2}}{2 m} \sqrt{\frac{2}{Π}} a^{3 / 2} \frac{a^{2} - 3 x^{2}}{{(x^{2} + a^{2})}^{3}}$}

_{$= \frac{h^{2}}{2 m} \sqrt{\frac{2}{Π} a^{3 / 2}}$ $\frac{1}{x^{2} + a^{2}} 2 \frac{a^{2} - 3 x^{2}}{{(x^{2} + a^{2})}^{2}}$}

The double derivative is calculated separately below:

_{$\frac{d^{2}}{{dx}^{2}} \frac{- 1}{x^{2} + a^{2}} = \frac{d}{dx} \frac{2 x}{{(x^{2} + a^{2})}^{2}}$}

$\frac{{(x^{2} + a^{2})}^{2} - x 2 (x^{2} + a^{2}) 2 x}{{(x^{2} + a^{2})}^{2}} = 2 \frac{a^{2} - 3 x^{2}}{{(x^{2} + a^{2})}^{3}}$

Therefore, the potential energy is $U = \frac{h^{2}}{m} \frac{3 x^{2} - a^{2}}{m {(x^{2} + a^{2})}^{2}}$ .

Sketch of potential energy and wave function

(b) The sketch is as follows. The blue line represents the potential energy, and the red line represents the wave function.

The unit length is

Find the region

The classical region is where the potential energy is smaller than 0, which is about $(- 0.6 a, 0.6 a)$ .

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a) Taking the particle’s total energy to be 0, find the potential energy.
(b) On the same axes, sketch the wave function and the potential energy.
(c) To what region would the particle be restricted classically?

Short Answer

Step by step solution

Potential energy

Sketch of potential energy and wave function

Find the region

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a) Taking the particle’s total energy to be 0, find the potential energy.(b) On the same axes, sketch the wave function and the potential energy.(c) To what region would the particle be restricted classically?

Short Answer

Step by step solution

Potential energy

Sketch of potential energy and wave function

Find the region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Further Mechanics and Thermal Physics

Dynamics

Electrostatics

Solid State Physics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

a) Taking the particle’s total energy to be 0, find the potential energy.
(b) On the same axes, sketch the wave function and the potential energy.
(c) To what region would the particle be restricted classically?