Calculate the energy difference between the ground state and the first excited state for an electron in one dimensional rigid box of length 10-8 cm.Also find the first three eigen functions.

To calculate the energy difference between the ground state and the first excited state for an electron in a one-dimensional rigid box, we can use the principles of quantum mechanics. The energy levels for a particle in a box are quantized, and the formula for the energy levels is given by:

Energy Levels in a One-Dimensional Box

The energy levels \(E_n\) for an electron in a one-dimensional box of length \(L\) are expressed as:

E_n = \frac{n^2 h^2}{8mL^2}

Where:

• E_n = energy of the nth level
• n = principal quantum number (1, 2, 3, ...)
• h = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
• m = mass of the electron (\(9.11 \times 10^{-31} \, \text{kg}\))
• L = length of the box (in this case, \(10^{-8} \, \text{cm} = 10^{-10} \, \text{m}\))

Calculating the Energy Levels

First, we need to convert the length of the box into meters:

L = \(10^{-8} \, \text{cm} = 10^{-10} \, \text{m}\)

Now, we can calculate the energy for the ground state (n=1) and the first excited state (n=2).

Ground State Energy (n=1)

Substituting \(n = 1\) into the energy formula:

E_1 = \frac{1^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot (9.11 \times 10^{-31}) \cdot (10^{-10})^2}

Calculating this gives:

E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \cdot (9.11 \times 10^{-31}) \cdot (10^{-20})} \approx 1.51 \times 10^{-18} \, \text{J}

First Excited State Energy (n=2)

Now for the first excited state (n=2):

E_2 = \frac{2^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot (9.11 \times 10^{-31}) \cdot (10^{-10})^2}

Calculating this gives:

E_2 = \frac{4 \cdot (6.626 \times 10^{-34})^2}{8 \cdot (9.11 \times 10^{-31}) \cdot (10^{-20})} \approx 6.04 \times 10^{-18} \, \text{J}

Energy Difference

The energy difference between the first excited state and the ground state is:

ΔE = E_2 - E_1

ΔE = (6.04 \times 10^{-18}) - (1.51 \times 10^{-18}) \approx 4.53 \times 10^{-18} \, \text{J}

Eigenfunctions of the System

The eigenfunctions for a particle in a one-dimensional box are given by:

ψ_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)

For the first three eigenfunctions:

First Eigenfunction (n=1)

ψ_1(x) = \sqrt{\frac{2}{10^{-10}}} \sin\left(\frac{\pi x}{10^{-10}}\right)

Second Eigenfunction (n=2)

ψ_2(x) = \sqrt{\frac{2}{10^{-10}}} \sin\left(\frac{2\pi x}{10^{-10}}\right)

Third Eigenfunction (n=3)

ψ_3(x) = \sqrt{\frac{2}{10^{-10}}} \sin\left(\frac{3\pi x}{10^{-10}}\right)

These eigenfunctions describe the probability distribution of finding the electron at a position \(x\) within the box. Each function corresponds to a specific energy level, with the ground state having the lowest energy and the higher states having progressively higher energies.

In summary, the energy difference between the ground state and the first excited state is approximately \(4.53 \times 10^{-18} \, \text{J}\), and the first three eigenfunctions are defined as shown above. This framework illustrates the fascinating behavior of quantum particles in confined spaces.