Chapter 2: Q19P (page 66)

Question: Find the probability current, $J$ (Problem 1.14) for the free particle wave function Equation $2.94$ . Which direction does the probability flow?

Short Answer

Expert verified

The probability current $(J)$ is

j = \frac{\hat{ℏ} k}{m} | A |^{2}

Step by step solution

The probability current density (J)

The required formula of probability current density $(J)$ is,

role="math" localid="1657786971189"

J = \frac{i \cap}{2 m} (Ψ \frac{\partial Ψ^{'}}{\partial x} - Ψ^{*} \frac{\partial Ψ}{\partial x})

Where $y$ is the wave function of the particle and complex conjugate of $y^{is} y °$ .

Compute probability current density

Equation $2.94$ from the chapter is

$Ψ_{k} (X, t) = A e^{i k x^{j k k^{2}}} 2 m^{2}$

ψ^{'} = A e^{i k \frac{k m^{2}}{2 m^{2}} ?},

Thus, it can written that,

$J = \frac{i ↻}{2 m} (Ψ \frac{\partial Ψ^{*}}{\partial x} - Ψ^{*} \frac{\partial Ψ}{\partial x})$

$= \frac{i \times}{2 m} | A |^{2} [e^{i (k x - \frac{x k^{2}}{2 m} t} (- i k) e^{- i (k x - \frac{a k^{2}}{2 m} t} - e^{(\{k x \frac{x k^{2} t}{2 m} t}) (i k) e^{(k x - \frac{m k^{2}}{2 m} t)}]$

$J = \frac{i ↻}{2 m} | A |^{2} (- 2 i k)$

$J = \frac{≧ k}{m} | A |^{2}$

It flows in the positive $(x)$ direction.

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Question: Find the probability current, $J$ (Problem 1.14) for the free particle wave function Equation $2.94$ . Which direction does the probability flow?

Short Answer

Step by step solution

The probability current density (J)

Compute probability current density

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Question: Find the probability current, J (Problem 1.14) for the free particle wave function Equation 2.94. Which direction does the probability flow?

Short Answer

Step by step solution

The probability current density (J)

Compute probability current density

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Astrophysics

Fields in Physics

Linear Momentum

Torque and Rotational Motion

Energy Physics

Study anywhere. Anytime. Across all devices.

Question: Find the probability current, $J$ (Problem 1.14) for the free particle wave function Equation $2.94$ . Which direction does the probability flow?