Chapter 5: Q95CE (page 193)

A bound particle of massdescribed by the wave function
${ψ(x)=Axe}^{{-x}^{2} {/2b}^{2}}$
What is the most probable location at which to find the particle?

Short Answer

Expert verified

The most probably location to find the particle is at. $x = \pm b$

Step by step solution

Wave function at maximum.

The wave function is

$ψ (x) = A x e^{- x^{2 / 2 b^{2}}}$

The most probably location to find the particle is when the probability density reach maximum,

$\begin{array}{l} ρ = ψ (x) ψ^{.} (x) \\ = {(A^{2} x^{2} e^{- x^{2 / b^{2}}})}^{2} \\ = A^{2} x^{2} e^{- x^{2 / b^{2}}} \end{array}$

Derivative of the density.

The derivative of the density

$\begin{matrix} \frac{d ρ}{d x} = A^{2} (2 x e^{- x^{2 / b^{2}}} + x^{2} e^{- x^{2 / b^{2}}} (- \frac{2 x}{b^{2}})) \\ = A^{2} 2 x e^{- x^{2 / b^{2}}} (1 - \frac{x^{2}}{b^{2}}) \end{matrix}$

When it is zero, the probability is maximum or minimum,

$A^{2} 2 x e^{- x^{2 / b^{2}}} (1 - \frac{x^{2}}{b^{2}}) = 0$ The solution is

$x = \pm b$ or $0$ .

At $x = 0$ , the density is $0$ , so it is a minimum.

Therefore, The most probably location to find the particle is at $x = \pm b$ .

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.

A bound particle of massdescribed by the wave function
${ψ(x)=Axe}^{{-x}^{2} {/2b}^{2}}$
What is the most probable location at which to find the particle?

Short Answer

Step by step solution

Wave function at maximum.

Derivative of the density.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Turning Points in Physics

Physics of Motion

Torque and Rotational Motion

Physical Quantities And Units

Waves Physics

Study anywhere. Anytime. Across all devices.

A bound particle of massdescribed by the wave functionψ(x)=Axe-x2/2b2What is the most probable location at which to find the particle?

Short Answer

Step by step solution

Wave function at maximum.

Derivative of the density.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Turning Points in Physics

Physics of Motion

Torque and Rotational Motion

Physical Quantities And Units

Waves Physics

Study anywhere. Anytime. Across all devices.

A bound particle of massdescribed by the wave function
${ψ(x)=Axe}^{{-x}^{2} {/2b}^{2}}$
What is the most probable location at which to find the particle?