Q7P Calculate d銆坧銆�/dt. Answer:d... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 1: Q7P (page 18)

Calculate d銆坧銆�/dt. Answer:
$\frac{d }{d x} = <- \frac{鈭� V}{鈭� x}>$

This is an instance of Ehrenfest鈥檚 theorem, which asserts that expectation values obey the classical laws

Short Answer

Expert verified

$\frac{d }{d x} = <- \frac{鈭� V}{鈭� x}>$

Schrodinger equation and its complex conjugate:

role="math" localid="1657778797745" $\frac{鈭� 蠄}{鈭� t} = \frac{i h}{2 m} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} - \frac{i}{h} V (x, t) 蠄 (x, t) (1) \frac{鈭� 蠄^{*}}{鈭� t} = - \frac{i h}{2 m} \frac{{鈭�}^{2} 蠄^{*}}{鈭� x^{2}} + \frac{i}{h} V (x, t) 蠄 (x, t) (2)$

And according to Ehrenfest鈥檚 theorem,

$ = m <v> = m \frac{d <x>}{d t}$

Step by step solution

Determining the expectation value of x

Finding the expectation values.

$<x> = \frac{{鈭�}_{- 鈭�}^{鈭�} x {|蠄_{(x . t)}|}^{2} d x}{{鈭�}_{- 鈭�}^{鈭�} {|蠄_{(x . t)}|}^{2} d x} <x> = {鈭�}_{- 鈭�}^{鈭�} x {|蠄_{(x . t)}|}^{2} d x <x> = {鈭�}_{- 鈭�}^{鈭�} 蠄_{(x . t)} {蠄^{*}}_{(x . t)} d x <x> = {鈭�}_{- 鈭�}^{鈭�} 蠄_{(x . t)} {蠄^{*}}_{(x . t)} d x$

Differentiating both the sides with respect to t,

$\frac{d (x)}{d t} = {鈭�}_{- 鈭�}^{鈭�} 蠄_{(x . t)} {蠄 {(x)}^{*}}_{(x . t)} d x \frac{d (x)}{d t} = {鈭�}_{- 鈭�}^{鈭�} 脳 \frac{鈭� 蠄}{鈭� t} [蠄_{(x . t)} {蠄^{*}}_{(x . t)}] d x \frac{d (x)}{d t} = {鈭�}_{- 鈭�}^{鈭�} 脳 (\frac{鈭� 蠄}{鈭� t} 蠄 + 蠄^{*} \frac{鈭� 蠄}{鈭� t}) d x$

Now, substituting the Schrodinger equation for the time derivatives

${鈭�}_{- 鈭�}^{鈭�} X (- \frac{i h}{2 m} \frac{{鈭�}^{2} 蠄^{*}}{鈭� x} 蠄 + \frac{i}{h} V 蠄^{*} 蠄 + \frac{i h}{2 m} 蠄^{*} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} - \frac{i}{h} V 蠄^{*} 蠄) d x$

$\frac{d <x>}{d t} = \frac{i h}{2 m} {鈭�}_{- 鈭�}^{鈭�} 脳 (蠄^{*} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} - \frac{{鈭�}^{2} 蠄^{*}}{鈭� x^{2}} 蠄) d x$

$\frac{d <x>}{d t} = \frac{i h}{2 m} {鈭�}_{- 鈭�}^{鈭�} 脳 [(\frac{鈭� 蠄^{*}}{鈭� x} \frac{鈭� 蠄}{鈭� x^{2}} + 蠄^{*} \frac{{鈭�}^{2} 蠄^{2}}{鈭� x^{2}}) - (\frac{{鈭�}^{2} 蠄^{2}}{鈭� x^{2}} 蠄 + \frac{鈭� 蠄^{*}}{鈭� x} \frac{鈭� 蠄}{鈭� x})] d x$

role="math" localid="1657786380532" $\frac{d <x>}{d t} = \frac{i h}{2 m} {鈭�}_{- 鈭�}^{鈭�} 脳 [\frac{鈭�}{鈭� x} (蠄^{*} \frac{鈭� 蠄}{鈭� x}) - \frac{鈭�}{鈭� x} (\frac{鈭� 蠄^{*}}{鈭� x} 蠄)] d x$

$\frac{d <x>}{d t} = \frac{i h}{2 m} {鈭�}_{- 鈭�}^{鈭�} 脳 [\frac{鈭�}{鈭� x} (蠄^{*} \frac{鈭� 蠄}{鈭� x}) - \frac{鈭�}{鈭� x} (\frac{鈭� 蠄^{*}}{鈭� x} 蠄)] d x \frac{d <x>}{d t} = \frac{i h}{2 m} 脳 (蠄^{*} 蠄_{- 鈭�}^{鈭�} - {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} \frac{鈭� 蠄}{鈭� x} d x - {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} \frac{鈭� 蠄}{鈭� x} d x) \frac{d <x>}{d t} = \frac{i h}{2 m} {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} \frac{鈭� 蠄}{鈭� x} d x$

Now, multiplying both sides by m

$m \frac{d <x>}{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} \frac{鈭� 蠄}{鈭� x} d x$

And using equation (3)

$ = - i h {鈭�}_{- 鈭�}^{鈭�} {蠄^{*}}^{鈭�} \frac{鈭� 蠄}{鈭� x} d x = {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} (- i h \frac{鈭�}{鈭� x}) 蠄 d x$

Differentiating both the sides with respect to t

We get the desired value,

\frac{d <p>}{d t} = - i h \frac{d}{d t} {鈭�}_{- 鈭�}^{鈭�} 蠄^{*} \frac{鈭� 蠄}{鈭� x} d x \frac{d <p>}{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} (蠄^{*} \frac{鈭� 蠄}{鈭� x}) d x

Using Clairaut’s theorem and substituting equation (1) and (2)

We get,

$\frac{d }{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} [(\frac{鈭� 蠄^{*}}{鈭� t}) \frac{鈭� 蠄}{鈭� t} + 蠄^{*} \frac{鈭�}{鈭� x} (\frac{鈭� 蠄}{鈭� t})] d x \frac{d }{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} [(- \frac{i h}{2 m} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} + \frac{i}{h} V 蠄^{*}) \frac{鈭� 蠄}{鈭� x} + 蠄^{*} \frac{鈭�}{鈭� x} (\frac{i h}{2 m} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} + \frac{i}{h} V 蠄)] d x$
$\frac{d }{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} [- \frac{i h}{2 m} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} \frac{鈭� 蠄}{鈭� x} + \frac{i}{h} V 蠄^{*} \frac{鈭� 蠄}{鈭� x} \frac{i h}{2 m} 蠄^{*} \frac{{鈭�}^{3} 蠄}{鈭� x^{3}} - \frac{i}{h} \frac{鈭� V}{鈭� x} 蠄^{*} 蠄 - \frac{i}{h} V 蠄^{*} \frac{鈭� 蠄}{鈭� x}] \frac{d }{d t} = - i h - (- \frac{i h}{2 m} \frac{鈭� 蠄^{*}}{鈭� x} \frac{鈭� 蠄}{鈭� x} - {鈭�}_{- 鈭�}^{鈭�} \frac{鈭� 蠄}{鈭� x} \frac{{鈭�}^{2} 蠄}{鈭� x^{2}} d x) + {鈭�}_{- 鈭�}^{鈭�} [\frac{i h}{2 m} 蠄 \frac{{鈭�}^{3} 蠄}{鈭� x^{3}} - \frac{i}{h} \frac{鈭� V}{鈭� x} 蠄^{*} 蠄] d x$

localid="1657947924145" $\frac{d }{d t} = - i h {鈭�}_{- 鈭�}^{鈭�} [- \frac{i h}{2 m} 蠄^{*} \frac{{鈭�}^{3} 蠄}{鈭� x} + \frac{i h}{2 m} 蠄^{*} \frac{{鈭�}^{3}}{鈭� x^{3}} - \frac{i}{h} \frac{鈭� V}{鈭� X} 蠄^{*} 蠄] d x \frac{d }{d t} = i^{2} {鈭�}_{- 鈭�}^{鈭�} [\frac{鈭� V}{鈭� X} 蠄^{*} 蠄] d x$

$\frac{d }{d t} = i^{2} {鈭�}_{- 鈭�}^{鈭�} [\frac{鈭� V}{鈭� x} 蠄^{*} 蠄] d x$

$\frac{d }{d t} = - {鈭�}_{- 鈭�}^{鈭�} [\frac{鈭� v}{鈭� x} 蠄^{*} 蠄] d x \frac{d }{d t} = {鈭�}_{- 鈭�}^{鈭�} [蠄^{*} (- \frac{鈭� V}{鈭� x}) 蠄] d x \frac{d }{d t} = <- \frac{鈭� V}{鈭� x}>$

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91影视

Calculate d銆坧銆�/dt. Answer:
$\frac{d <p>}{d x} = <- \frac{鈭� V}{鈭� x}>$

This is an instance of Ehrenfest鈥檚 theorem, which asserts that expectation values obey the classical laws

Short Answer

Step by step solution

Determining the expectation value of x

Differentiating both the sides with respect to t,

Now, substituting the Schrodinger equation for the time derivatives

Differentiating both the sides with respect to t

Using Clairaut’s theorem and substituting equation (1) and (2)

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Most popular questions from this chapter

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91影视

Calculate d銆坧銆�/dt. Answer:dpdx=-鈭�V鈭�x This is an instance of Ehrenfest鈥檚 theorem, which asserts that expectation values obey the classical laws

Short Answer

Step by step solution

Determining the expectation value of x

Differentiating both the sides with respect to t,

Now, substituting the Schrodinger equation for the time derivatives

Differentiating both the sides with respect to t

Using Clairaut’s theorem and substituting equation (1) and (2)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Turning Points in Physics

Radiation

Famous Physicists

Conservation of Energy and Momentum

Thermodynamics

Study anywhere. Anytime. Across all devices.

Calculate d銆坧銆�/dt. Answer:
$\frac{d <p>}{d x} = <- \frac{鈭� V}{鈭� x}>$

This is an instance of Ehrenfest鈥檚 theorem, which asserts that expectation values obey the classical laws