When a particle is restricted to move along x-axis between `x=0` and `x=a`, where `alpha` if of nenometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends `x=0` and `x=a`. The wavelength of this standing wave is related to the linear momentum p of the particle according to the de Broglie relation. The energy of the particle of mass m is related to its linear momentum as `E=(p^2)/(2m)`. Thus the energy of the particle can be denoted by a quantum number `n` taking values 1,2,3, ...(`n=1`, called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving along the line from `x=0` to `x=alpha`. Take `h=6.6xx10^(-34)Js` and `e=1.6xx10^(-19)` C.
Q. The allowed energy for the particle for a particular value of n is proportional to

Options

A. `alpha^(-2)`
B. `alpha^((-3)/(2))`
C. `alpha^(-1)`
D. `alpha^2`

Explanation

To find the allowed energy for a particle moving along the x-axis between \( x = 0 \) and \( x = \alpha \), we can follow these steps:

### Step 1: Understanding the standing wave condition
The particle is confined between \( x = 0 \) and \( x = \alpha \). The allowed energies correspond to standing waves with nodes at the boundaries. The wavelength \( \lambda \) of the standing wave is related to the length of the box by:
\[ \alpha = n \frac{\lambda}{2} \]
where \( n \) is a positive integer (quantum number).
...