Chapter 2: Q9P (page 40)

For the wave function in Example 2.2, find the expectation value of H, at time $t = 0$ ,the “old fashioned way:
$⟨ H ⟩ = \int Ψ {(x, 0)}^{*} \hat{H} Ψ (x, 0) dx .$
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" $\frac{5 ℏ^{2}}{{ma}^{2}}$ 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 Ψ 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 $\hat{H} Ψ (x, 0)$ have to be found for the calculation of expectation value of H:

$\begin{matrix} \hat{H} Ψ (x, 0) = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} [Ax (a - x] \\ = - A \frac{ℏ^{2}}{2 m} \frac{\partial}{\partial x} (a - 2 x) \\ = A \frac{ℏ^{2}}{m} \end{matrix}$

Finding the value of H

The expectation value of H can be calculated as,

$\begin{matrix} ⟨ H ⟩ = \int Ψ (x, 0) \hat{H} Ψ (x, 0) dx \\ = A^{2} \frac{ℏ^{2}}{m} \int_{0}^{a} x (a - x) dx \\ = {A^{2} \frac{ℏ^{2}}{m} (a \frac{x^{2}}{2} - \frac{x^{3}}{3}) |}_{0}^{a} \\ = A^{2} \frac{ℏ^{2}}{m} (\frac{a^{2}}{2} - \frac{a^{3}}{3}) \\ = \frac{30}{a^{5}} \frac{ℏ^{2}}{m} \frac{a^{3}}{6} A \\ = = \frac{5 ℏ^{2}}{{ma}^{2}} \end{matrix}$

Thus, the expectation value of $H$ is $\frac{5 ℏ^{2}}{{ma}^{2}}$ 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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For the wave function in Example 2.2, find the expectation value of H, at time t=0 ,the “old fashioned way:⟨H⟩=∫Ψ(x,0)∗H^Ψ(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

Classical Mechanics

Radiation

Medical Physics

Electric Charge Field and Potential

Mechanics and Materials

Fluids

Study anywhere. Anytime. Across all devices.