Question

A proton confined in a one-dimensional box emits a 2.0 MeV gamma-ray photon in a quantum jump What is the length of the box? from n=2 to n=1. Express your answer in meters.

Answer 1

To find the length of the box, we can use the equation E = ((n^2 * h^2) / (8 * m * L^2)), and rearrange it to solve for L. Plugging in the values for a proton and a 2.0 MeV energy jump, we can calculate the length of the box in meters.

Step-by-step explanation:

The energy levels of a particle confined in a one-dimensional box are given by the equation:

E = ((n^2 * h^2) / (8 * m * L^2)),

where E is the energy level, n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

To find the length of the box, we can rearrange the equation: L = ((n^2 * h^2) / (8 * m * E))^(1/2),

where n = 1 and E = 2.0 MeV = 2.0 * 10^6 eV. So, plugging in the values, we get: L = ((1^2 * h^2) / (8 * m * (2.0 * 10^6)))^(1/2).

Let's solve for L. The mass of a proton (m) is 1.67 * 10^(-27) kg, and Planck's constant (h) is 6.63 * 10^(-34) J*s. Plugging in these values, we get:

L = ((1^2 * (6.63 * 10^(-34))^2) / (8 * (1.67 * 10^(-27)) * (2.0 * 10^6))))^(1/2).

Solving this equation gives us the length of the box in meters.

Answer 2

The length of the one-dimensional box in which the proton is confined can be determined using the energy formula for a particle in a box. The given energy of the emitted gamma-ray photon, 2.0 MeV, corresponds to the energy difference between quantum states ( n = 2 ) and ( n = 1 ). This energy difference is then used in the formula to find the length of the box.

Step-by-step explanation:

The energy levels of a particle in a one-dimensional box are quantized, and the energy difference between these levels is given by the equation:

`\[ \Delta E = \frac{{n^2h^2}}{{8mL^2}} \]`

Here,
`\( \Delta E \)` is the energy difference, ( n ) is the quantum number, ( h ) is the Planck constant, ( m ) is the mass of the particle, and ( L ) is the length of the box.

Given that the proton emits a 2.0 MeV gamma-ray photon from ( n = 2 ) to ( n = 1 ), the energy difference
`\( \Delta E \)` can be calculated using the energy formula
`\( E = mc^2 \)`. Substituting
`\( \Delta E \)` into the energy difference formula and solving for ( L ) yields the length of the box.

This approach provides a method to determine the size of the confinement region for the proton based on the observed energy transition. The calculation ensures that the length of the one-dimensional box is consistent with the energy of the emitted gamma-ray photon.