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 $x = \pm b$ .

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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

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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

Electromagnetism

Fluids

Translational Dynamics

Conservation of Energy and Momentum

Modern Physics

Fundamentals of 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?