Q9P For the wave function in Example... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 2: Q9P (page 40)

For the wave function in Example 2.2, find the expectation value of H, at time $t = 0$ ,the 鈥渙ld fashioned way:
$鉄� H 鉄� = 鈭� 唯 {(x, 0)}^{鈭�} \hat{H} 唯 (x, 0) dx .$
Compare the result obtained in Example 2.3. Note: Because $<H>$ is independent of time, there is no loss of generality in using $t = 0$

Short Answer

Expert verified

The expectation value of $H$ isrole="math" localid="1655393475453" $\frac{5 {鈩�}^{2}}{{ma}^{2}}$ which is same as Example .

Step by step solution

Definition of wave function

A wave function is a function that describes the probability of a particle's quantum state as a function of position, momentum, time or spin. The variable 唯 is widely used to represent wave functions.

Finding the value of H^Ψ(x,0)

The expectation value of $H$ at the time $t = 0$ will be evaluated.

The value of $\hat{H} 唯 (x, 0)$ have to be found for the calculation of expectation value of H:

$\begin{matrix} \hat{H} 唯 (x, 0) = 鈭� \frac{{鈩�}^{2}}{2 m} \frac{{鈭�}^{2}}{鈭� x^{2}} [Ax (a 鈭� x] \\ = 鈭� A \frac{{鈩�}^{2}}{2 m} \frac{鈭�}{鈭� x} (a 鈭� 2 x) \\ = A \frac{{鈩�}^{2}}{m} \end{matrix}$

Finding the value of H

The expectation value of H can be calculated as,

$\begin{matrix} 鉄� H 鉄� = 鈭� 唯 (x, 0) \hat{H} 唯 (x, 0) dx \\ = A^{2} \frac{{鈩�}^{2}}{m} {鈭�}_{0}^{a} x (a 鈭� x) dx \\ = {A^{2} \frac{{鈩�}^{2}}{m} (a \frac{x^{2}}{2} 鈭� \frac{x^{3}}{3}) |}_{0}^{a} \\ = A^{2} \frac{{鈩�}^{2}}{m} (\frac{a^{2}}{2} 鈭� \frac{a^{3}}{3}) \\ = \frac{30}{a^{5}} \frac{{鈩�}^{2}}{m} \frac{a^{3}}{6} A \\ = = \frac{5 {鈩�}^{2}}{{ma}^{2}} \end{matrix}$

Thus, the expectation value of $H$ is $\frac{5 {鈩�}^{2}}{{ma}^{2}}$ which is same as Example $2.3$ .

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Short Answer

Step by step solution

Definition of wave function

Finding the value of H^Ψ(x,0)

Finding the value of H

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91影视

For the wave function in Example 2.2, find the expectation value of H, at time t=0 ,the 鈥渙ld fashioned way:鉄�H鉄�=鈭�唯(x,0)鈭�H^唯(x,0)dx.Compare the result obtained in Example 2.3. Note: BecauseH is independent of time, there is no loss of generality in usingt=0

Short Answer

Step by step solution

Definition of wave function

Finding the value of H^Ψ(x,0)

Finding the value of H

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Fluids

Further Mechanics and Thermal Physics

Radiation

Astrophysics

Electricity

Study anywhere. Anytime. Across all devices.