Calculation of the Length of the Box for a Ruby Laser

Given:
- Wavelength of the emitted light (λ) = 694.3 nm
- Transition of the electron in a box from n=2 to n=1 state

To find the length of the box, we can use the formula for the allowed energy levels of a particle in a one-dimensional box:

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

Where:
- E is the energy of the particle
- n is the principal quantum number
- h is the Planck's constant (6.626 x 10^-34 J.s)
- m is the mass of the particle
- L is the length of the box

We know that the energy difference between two levels is given by:

ΔE = E2 - E1 = ((n2^2 * h^2) / (8 * m * L^2)) - ((n1^2 * h^2) / (8 * m * L^2))

Given that the wavelength of the emitted light is related to the energy difference by the equation:

ΔE = hc / λ

Where:
- h is the Planck's constant
- c is the speed of light
- λ is the wavelength

Solving for L:
Let's substitute the values and solve for L:

((n2^2 * h^2) / (8 * m * L^2)) - ((n1^2 * h^2) / (8 * m * L^2)) = hc / λ

Simplifying the equation:

(n2^2 - n1^2) * h^2 / (8 * m * L^2) = hc / λ

(n2^2 - n1^2) * h^2 = 8 * m * L^2 * hc / λ

L = √((n2^2 - n1^2) * h^2 * λ / (8 * m * c))

Substituting the given values:
- n2 = 2
- n1 = 1
- h = 6.626 x 10^-34 J.s
- λ = 694.3 nm = 694.3 x 10^-9 m
- m (mass of the electron) = 9.109 x 10^-31 kg
- c (speed of light) = 3 x 10^8 m/s

Calculating the length:

L = √((2^2 - 1^2) * (6.626 x 10^-34 J.s)^2 * (694.3 x 10^-9 m) / (8 * (9.109 x 10^-31 kg) * (3 x 10^8 m/s)))

L = √((3.50 x 10^-68 J^2.s^2.m) / (2.909 x 10^-22 kg.m/s))

L = √(1.20 x 10^-46 m^2)

L ≈ 1.10 x 10^-23 m

Therefore, the length of the box for the given Ruby laser is approximately 1.10 x 10^-23