Obtain the eigen values and normalized wave functions for a particle in a one dimensional infinite potential box of side " \(a "\) ".

The Schrödinger equation is a fundamental equation in quantum mechanics. It describes how the quantum state of a physical system changes over time. For a particle in a one-dimensional infinite potential box, we use the time-independent Schrödinger equation:
[\[\begin{equation} -\frac{abla^2 \text{ψ}}{2m} + V(x)\text{ψ} = E\text{ψ} onumber \tag{1} \end{equation}\]] Since the potential V(x) is zero inside the box (0 < x < a) and infinite outside, the equation simplifies:
[\[\begin{equation} -\frac{abla^2 \text{ψ}}{2m} = E\text{ψ} onumber \tag{2} \end{equation}\]] This simplified form directly relates the kinetic energy of the particle (left side) to its total energy E (right side). Solving this differential equation helps us understand the allowable energy levels (eigenvalues) and the corresponding wave functions of the particle.

Eigenvalues are critical when solving the Schrödinger equation. They represent the allowed energy levels of the particle in the box. To find eigenvalues, we transform the simplified Schrödinger equation:
[\[\begin{equation} \frac{d^2 \text{ψ}}{dx^2} + \frac{2mE}{abla^2}\text{ψ} = 0 onumber \tag{3} \end{equation}\]] Setting \text{k^2 = \frac{2mE}{abla^2}}, it becomes:
[\[\begin{equation} \frac{d^2 \text{ψ}}{dx^2} + k^2\text{ψ} = 0 onumber \tag{4} \end{equation}\]] The solutions to this differential equation are:
[\[\begin{equation} \text{ψ}(x) = A \text{sin}(kx) + B \text{cos}(kx) onumber \end{equation}\]] Applying boundary conditions allows us to solve for k and find the allowed energy levels:
[\[\begin{equation} k = \frac{n\theta}{a}\text{, } n = 1, 2, 3, ... onumber \tag{5} \end{equation}\]] Thus, the energy eigenvalues are:
[\[\begin{equation} E_n = \frac{n^2 \theta^2}{2ma^2} onumber \tag{6} \end{equation}\]]. Each eigenvalue represents a quantized energy level that the particle can occupy.

Wave functions describe the quantum state of a particle. For our infinite potential box, they are solutions to the Schrödinger equation. The general form of the wave function, after applying boundary conditions, is:
[\[\begin{equation} \text{ψ}(x) = A \text{sin}(kx) onumber \tag{7} \end{equation}\]] Boundary conditions help us determine that:
[\[\begin{equation} k = \frac{n \theta}{a}\text{, } n = 1, 2, 3, ... onumber \tag{8} \end{equation}\]] Thus, the wave functions for the nth state are:
[\[\begin{equation} \text{ ψ}_n (x) = A \text{sin} (\frac{n\theta x}{a}) onumber \end{equation}\]] This solution implies that we have standing waves within the box, where \( n \) determines the number of nodes (points where ψ(x) = 0) inside the box. Higher \( n \) values correspond to higher-energy states with more nodes.

Boundary conditions are crucial for solving the Schrödinger equation within the infinite potential box. They ensure that the wave function satisfies the physical constraints of the problem. For our box, the wave function must be zero at the boundaries (x=0 and x=a) because the potential outside the box is infinite. Thus, we have:
[\[\begin{equation}\text{ ψ}(0) = 0 onumber \tag{9} \end{equation}\]] and:
[\[\begin{equation}\text{ ψ}(a) = 0 \tag{10} \end{equation}\]] Applying these conditions to the general solution:
[\[\begin{equation}\text{ A sin}(kx) + B \text{cos}(kx) onumber \end{equation}\]] allows for nullifying the cosine term (setting B = 0) and establishing:
[\[\begin{equation} \text{ψ}(x) = A \text{sin}(kx)onumber \end{equation}\]] Then applying:
[\[\begin{equation}\text{ψ}(a)=0 \tag{11} \end{equation}\]] gives us our quantized values for k. These conditions are essential to finding valid eigenvalues and corresponding wave functions.

Normalization ensures that the total probability of finding the particle within the box is 1. This means that the integral of the square of the wave function over the entire box must be equal to 1:
<\[\begin{equation}\text{∫}_{0}^{a} |\text{ψ}(x)|^2 dx = 1 \tag{12} >onumber \end{equation}\]] For our normalized wave function:
[\[\begin{equation}\text{ψ}_n(x) = A \text{sin}(\frac{n C x}{a}) onumber \tag{13} \end{equation}\]]< \br> this implies:
<[\[\begin{equation}A = \frac{\text{sqrt(2)}}{a} onumber \tag{14} \end{equation}\]] Therefore, the normalized wave functions for any state n are:
[\[\begin{equation} \text{ψ}_n(x) = \frac{\text{sqrt}(2)}{a} \text{sin}(\frac{n C x}{a}) onumber \tag{15} \end{equation}\]] This ensures that the wave functions are properly scaled to represent the probabilities accurately. Proper normalization is fundamental in quantum mechanics to make meaningful physical predictions.