Calculating Energies in Box Potential

[Messy]

Homework Statement

Consider the box potential
V (x) = 0 0 < x < a
1 elsewhere

a) Estimate the energies of the ground state as well as those of the First and second
excited states for
i) an electron enclosed in a box of size a = 10E-10 m. Express your answer in
electron volts.
ii) a 1 g metallic sphere which is moving in box of size a = 10 cm. Express your
answer in joules.
b) Discuss the importance of quantum effects for both these two systems.

Homework Equations

E1 = ((h)^2 ) / ( 8 * m * L ^2)

En = n^2 * E1

The Attempt at a Solution

The First problem is , I am not sure about L(Length) . Is this equal to the size of the box 'a' in part i and ii.

The Second Problem is how can i define the importantance of quantum effect (b) .. I mean should i have to define with the varying size what change i am observing in Energy Levels .

 

[Messy]

May be i have to put this one in Advance Physics Sections :-)

 

[thomate1]

I would like to address the homework statement by first clarifying the parameters used in the equations provided. In this case, the length 'L' refers to the size of the box 'a' in both parts i and ii.

For part i, where an electron is enclosed in a box of size a = 10E-10 m, the energy levels can be estimated using the equations provided. The ground state energy (n=1) would be E1 = ((h)^2 ) / (8 * m * (10E-10)^2) = 3.5 x 10^-18 eV. Similarly, the first excited state (n=2) would be E2 = (2^2) * E1 = 4E1 = 1.4 x 10^-17 eV and the second excited state (n=3) would be E3 = (3^2) * E1 = 9E1 = 3.2 x 10^-17 eV.

For part ii, where a 1 g metallic sphere is moving in a box of size a = 10 cm, the energy levels can be estimated using the same equations, but the units need to be converted to joules. The ground state energy (n=1) would be E1 = ((h)^2 ) / (8 * m * (10^-2)^2) = 5.6 x 10^-25 J. Similarly, the first excited state (n=2) would be E2 = (2^2) * E1 = 4E1 = 2.2 x 10^-24 J and the second excited state (n=3) would be E3 = (3^2) * E1 = 9E1 = 5E1 = 1.7 x 10^-24 J.

Moving on to the discussion of the importance of quantum effects for both these systems, it is essential to understand that classical mechanics fails to explain the behavior of particles at the atomic and subatomic level. Quantum mechanics, on the other hand, provides a more accurate description of the behavior of these particles.

In the case of an electron enclosed in a box, quantum effects are crucial in determining the energy levels and the probability of finding the electron at a particular energy level. The size of the box plays a significant role in determining the energy levels, as seen in the equations