Q30P Normalize 蠄(x)聽the equation 2.... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 2: Q30P (page 83)

Normalize $蠄 (x)$ 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

$= 2 {鈭�}_{0}^{鈭�} | 蠄 |^{2} d x = 2 (| D |^{2} {鈭�}_{0}^{a} \cos^{2} l x + | F |^{2} {鈭�}_{0}^{鈭�} e^{鈭� 2 k x} d x)$

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

But $F = {De}^{ka} cosla (Eq 鈰� 2 .152)$ , 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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91影视

Normalize $蠄 (x)$ 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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91影视

Normalize 蠄(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

Electricity and Magnetism

Physics of Motion

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Study anywhere. Anytime. Across all devices.

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