Chapter 2: Q30P (page 83)

Normalize $ψ (x)$ the equation 2.151, to determine the constants D and F.

Short Answer

Expert verified

$D = \frac{1}{\sqrt{a + \frac{1}{k}}}$

$F = \frac{e^{ka} cosla}{\sqrt{a + \frac{1}{k}}}$

Step by step solution

Define the Schrödinger equation

A differential equation describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

Determine the Constants

$= 2 \int_{0}^{\infty} | ψ |^{2} d x = 2 (| D |^{2} \int_{0}^{a} \cos^{2} l x + | F |^{2} \int_{0}^{\infty} e^{- 2 k x} d x)$

$= 2 [{| D |^{2} (\frac{x}{2} + \frac{1}{4 l} \sin 2 l x) |}_{0}^{a} + {{| | F |}^{2} (- \frac{1}{2 k} e^{- 2 k x}) |}_{a}^{\infty} 2 [| D |^{2} (\frac{a}{2} + \frac{\sin 2 l a}{4 l}) + | F |^{2} \frac{e^{- 2 k a}}{2 k}]$

But $F = {De}^{ka} cosla (Eq \cdot 2 .152)$ , so role="math" localid="1656054797733" $1 = | D |^{2} (a + \frac{\sin (2 l a)}{2 l} + \frac{\cos^{2} (l a)}{k})$

Furthermore $k = l \tan (l a) (Eq .2154)$ , so

$= | D |^{2} (a + \frac{2 \sin l a \cos l a}{2 l} + \frac{\cos^{3} l a}{l \sin l a})$

$= | D |^{2} [a + \frac{\cos l a}{l \sin l a} (\sin^{2} l a + \cos^{2} l a)]$

$= | D |^{2} (a + \frac{1}{l \tan l a})$

$= | D |^{2} (a + \frac{1}{k})$

$D = \frac{1}{\sqrt{a + \frac{1}{k}}}$

$F = \frac{e^{k a} \cos l a}{\sqrt{a + \frac{1}{k}}}$

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Normalize $ψ (x)$ the equation 2.151, to determine the constants D and F.

Short Answer

Step by step solution

Define the Schrödinger equation

Determine the Constants

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Normalize ψ(x)the equation 2.151, to determine the constants D and F.

Short Answer

Step by step solution

Define the Schrödinger equation

Determine the Constants

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Torque and Rotational Motion

Modern Physics

Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.

Normalize $ψ (x)$ the equation 2.151, to determine the constants D and F.