Using de Broyglie’s equation $p = \frac{h}{\lambda}$ and $L = n \frac{\lambda_{n}}{2}$, the momentum of the particle in its n^{th} mode is given by:

$p_{n} = \frac{h}{\lambda_{n}} = \frac{nh}{2L}$

The momentum p is also related to the kinetic energy of the particle E_{k} by:

(To derive this formula, sub. p = mv into KE eqn)

$E_{k} = \frac{p^{2}}{2m}$

Since the potential energy V within the box is zero, the total energy E of the particle is equal to its kinetic energy E_{k}.

$E = E_{k} = \frac{p^{2}}{2m}$

Hence, using the above equations, the energy of the particle in its n^{th} mode is given by:

$E_{n} = \frac{p_{n}^{2}}{2m} = \frac{ \left( \frac{nh}{2L} \right)^{2}}{2m} = \frac{n^{2} h^{2}}{8m L^{2}}$

**Note:**

- The energy of the particle can only take discrete values(it is quantised). The number n, which identifies the quantum state of the particle, is called the quantum number. The energy increases as n increases.
- As n cannot be zero (explained below), the minimum energy is for n = 1, $E_{1} = \frac{h^{2}}{8mL^{2}}$. When confined to the box, the particle cannot have zero energy. It must have at least this minimum energy, called the zero point energy (or ground-state energy), ie. the confined particle can never be still.

The impossibility of the particle in the box having zero energy can be seen as a consequence of the uncertainty principle.

If the particle is at rest (ie. its energy is zero), then the particle’s momentum would also be zero, and this means that we would know its momentum precisely. In other words, the uncertainty Δp in the momentum would be zero. However, we know that the particle is confined to the box, and therefore the uncertainty Δx in its position is L. Hence, ΔpΔx = 0 which violates the uncertainty principle.

In the limit as L $\rightarrow$ ∞, the zero point E_{1} approaches zero. In this limit, with an infinitely wide box, the particle becomes a free particle (do not have potential energy), no longer confined in the x-direction. And because the energy of a free particle is not quantized, that energy can have any value, including zero.

Only a confined particle must have a finite zero point energy and can never be at rest.