solve the quantum physics particle in the box problem via schrodinger's equation

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The particle in a box problem is one of the most fundamental examples in quantum mechanics. We have a particle of mass m confined to move in a one-dimensional box of length L. The potential is zero inside the box and infinite at the walls, meaning the particle cannot escape. To solve this problem, we start with the time-independent Schrödinger equation. The Hamiltonian operator acts on the wavefunction to give the energy times the wavefunction. When we expand this, we get the kinetic energy term plus the potential energy term. Inside the box where the potential is zero, the equation simplifies significantly. We solve the differential equation by substituting k squared equals 2mE over h-bar squared. This gives us a standard harmonic oscillator equation with general solution involving sine and cosine functions. Applying the boundary conditions that the wavefunction must be zero at both walls eliminates the cosine term and constrains the allowed values of k. The boundary condition at x equals L requires that sine of kL equals zero, which means kL must equal n pi, where n is a positive integer. This quantizes the allowed values of k and therefore the energy. The energy levels are proportional to n squared, creating unequal spacing between levels. The lowest energy state, n equals 1, is called the ground state. The final step is normalization to ensure the total probability equals one. This gives us the complete solution: the normalized wavefunctions are square root of 2 over L times sine of n pi x over L, with quantized energy levels proportional to n squared. These standing wave patterns show nodes at the walls and demonstrate the fundamental quantum mechanical principle that energy is quantized in confined systems.