Chapter 7, Problem 7D.8P

(a)

Interpretation Introduction

Interpretation:

The reason for the probability density of finding a classical particle confined to move in a one dimensional box is $1/L$ 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.

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

(b)

Interpretation Introduction

Interpretation:

The reason for the average value of $\langle x^n\rangle$ for a particle of mass $m$ in a one dimensional box of length $L$ is ${\displaystyle \underset{0}{\overset{L}{\int }}P\left(x\right)x^n\text{d}x}$ 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.

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

(c)

Interpretation Introduction

Interpretation:

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

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 very large values of the quantum numbers, the predictions of quantum mechanics and classical mechanics are similar 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.

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