Chapter 28, Problem 71P

(a)

To determine

The sketches of the wave function of a particle confined in a finite box.

Answer to Problem 71P

The wave function is given below.

Explanation of Solution

For a particle in a box, the wave function in the $n^{\text{th}}$ state has $n-1$ nodes.

For $n=1$ state, the wave function is given below:

Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than half the wavelength. Thus,

$\frac{\lambda }{2}>L$ and $\lambda <\infty$

i.e.$\lambda >\frac{2L}{1}$ and $\lambda <\frac{2L}{1-1}$

Here, $\lambda$ is the wavelength and $L$ is the length of the box.

For $n=2$ state, the wave function is given below:

Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than one wavelength and more than half the wavelength.

$\lambda >L$ and $\lambda <2L$

i.e.$\lambda >\frac{2L}{2}$ and $\lambda <\frac{2L}{2-1}$

For $n=3$ state, the wave function is given below:

Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than $1.5$ times the wavelength but more than one wavelength.

$\frac{3\lambda }{2}>L$ and $\lambda <L$

i.e.$\lambda >\frac{3L}{2}$ and $\lambda <\frac{2L}{3-1}$

From all the above given wave functions, it can be generalised that

$\frac{2L}{n}<{\lambda }_n<\frac{2L}{n-1}$                                                             (I)

Here, ${\lambda }_n$ is the wave function of the $n^{\text{th}}$ state.

(b)

To determine

Check whether the condition ${\left(n-1\right)}^2{E}_1<E_n<n^2{E}_1$  is satisfied for a particle confined in a finite box.

Answer to Problem 71P

Yes. The given condition is valid for a particle confined in a finite box.

Explanation of Solution

The energy of $3d$ level is $-1.6\text{ eV}$ and the energy of $3p$ level is $-3.0\text{ eV}$.

Write the expression for energy of a particle in a infinite box in terms of wavelength

$E_n=\frac{h^2}{2m{\lambda }_n^2}$                                                                    (II)

Here, $E_n$ is the energy of the $n^{\text{th}}$ state, $h$ is the Planck’s constant and $m$ is the mass of the particle.

Rearranging (II) for ${\lambda }_n^2$

${\lambda }_n^2=\frac{h^2}{2m{E}_n}$                                                           (III)

Write the expression for ground state energy

$E_1=\frac{h^2}{8m{L}^2}$                                                           (IV)

Write expression (I)

$\frac{2L}{n}<{\lambda }_n<\frac{2L}{n-1}$                                                            (I)

Taking reciprocal and squaring (I)

$\frac{{\left(n-1\right)}^2}{{\left(2L\right)}^2}<\frac{1}{{\lambda }_n^2}<\frac{n^2}{{\left(2L\right)}^2}$                                                             (V)

While taking reciprocal, the direction of inequality changes.

Multiplying (III) by $E_1{\left(2L\right)}^2$

${\left(n-1\right)}^2{E}_1<\frac{{\left(2L\right)}^2{E}_1}{{\lambda }_n^2}<n^2{E}_1$                                                         (VI)

Substituting (III) in (VI)

${\left(n-1\right)}^2{E}_1<\frac{{\left(2L\right)}^2{E}_{1}2m{E}_n}{h^2}<n^2{E}_1$                                                          (VII)

Substituting for $E_1$ from (IV) in (VII)

${\left(n-1\right)}^2{E}_1<\left(\frac{h^2}{8m{L}^2}\right)\frac{8m{L}^2{E}_n}{h^2}<n^2{E}_1$

Cancelling out the common terms in numerator and denominator

${\left(n-1\right)}^2{E}_1<E_n<n^2{E}_1$

Thus, the given condition is valid.