Chapter 7, Problem 7D.6P

(a)

Interpretation Introduction

Interpretation:

The reason as to why the expectation value of position $\left(x\right)$ is $L/2$, for a particle of mass $m$ in a one dimensional box of length $L$, without the evaluation of any integral has to be stated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$

(b)

Interpretation Introduction

Interpretation:

The reason as to why the expectation value of momentum $\left(p_x\right)$ is $0$, for a particle of mass $m$ in a one dimensional box of length $L$, without the evaluation of any integral has to be given.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$

(c)

Interpretation Introduction

Interpretation:

The value of $\langle x^2\rangle$ for a particle of mass $m$ in a one dimensional box of length $L$ has to be calculated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$

(d)

Interpretation Introduction

Interpretation:

The reason for the value of $\langle p_x{}^2\rangle =n^2{h}^2/4L^2$ has to be explained.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by $\psi$ in the quantum mechanics.  The normalized wavefunction of a particle of mass $m$ in a one dimensional box of length $L$ for a general quantum state $n$ is shown below.

    $\psi ={\left(\frac{2}{L}\right)}^{1/2}\sin \left(n\pi x/L\right)$