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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Magnetism

Fields in Physics

Medical Physics

Particle Model of Matter

Space Physics

Wave Optics

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?

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

Magnetism

Fields in Physics

Medical Physics

Particle Model of Matter

Space Physics

Wave Optics

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?