Obtain an approximate analytic cxpression for the energy level in a square well potential \((l=0)\) when \(V_{0} a^{2}\) is slightly greater than \(\pi^{2} \hbar^{2} / 8 m\)

The Schrödinger equation is a fundamental part of quantum mechanics. It describes how the quantum state of a physical system changes over time. Here, we focus on the **time-independent Schrödinger equation** for a particle in a one-dimensional potential well, which is given by: \[ - \frac{\f hbar^2}{2m} \frac{\f d^2 \f psi}{\f dx^2} + \f V(x) \f psi = \f E \f psi \]. The term \( \frac{\f hbar^2}{2m} \frac{\f d^2 \f psi}{\f dx^2} \) represents the kinetic energy of the particle, \( \f V(x) \f psi \) represents the potential energy, and \( \f E \f psi \) represents the total energy of the system.
Understanding the Schrödinger equation helps us determine the allowed energy levels of a particle in a potential well.
To solve the Schrödinger equation for a square well potential, we start by defining the potential energy function \( \f V(x) \).

Quantization refers to the fact that certain physical properties, like energy, can only take on discrete values. In the context of a particle in a square well potential, the presence of boundaries at \( \f x = \f \fspm \f a \) imposes specific conditions on the wavefunction \( \f psi \).
The wavefunction must satisfy the boundary conditions: \( \f psi = 0 \) at \( \f x = \f \fsm \f a \). This requirement ensures that the wavefunction properly describes a confined particle and leads to quantized energy levels.
In our example, the potential inside the well is zero, and outside it is \( V_0 \). The quantization condition for a particle in the well can be approximated by considering that the product of the potential height \( V_0 \) and the width \( a \)**2 is only slightly greater than a critical value: \( V_0 a^2 \f \f approx\f \frac{\f \fpipr^2 \f hbar^2}{\f 8m} + \f epsilon \), where \( \f epsilon \) is a small correction factor.

The energy eigenvalue equation helps determine the specific energy levels (eigenvalues) that a quantum system can occupy. For a particle in a square well, the energy levels are derived from the Schrödinger equation and the quantization condition.
The energy levels inside a square well can be approximated using the formula:

• \( E_n \f approx \f \frac{n^2 \f psipr^2 \f hbar^2}{2m a^2} - \f \frac{\f epsilon}{a^2} \).

Here, \( n \) is an integer known as the quantum number, representing different energy levels, starting from \( n = 1 \) for the ground state.
For the ground state (when \( n=1 \)), the energy eigenvalue equation becomes:
\( E_1 \f approx \f \frac{\f psipr^2\f hbar^2}{2m a^2} - \f \frac{\f epsilon}{a^2} \).
This expression shows that the actual energy level is slightly lower than in the ideal case. Approximations like these help us understand the behavior of particles in potential wells under varying conditions.