Particle in a One-Dimensional Box
The Particle in a One-Dimensional Box is a simple quantum mechanical system used to illustrate basic principles of quantum mechanics. In this scenario, a particle is confined to move within a one-dimensional box of length \(L\).
Derivation for Normalized Wave Function:
The time-independent Schrödinger equation for this system is given by:
\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\]
For a particle in a box, \(V(x)\) is zero within the box and infinite outside the box. The solution to the Schrödinger equation within the box (\(0 < x < L\)) is given by:
\[\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)\]
The normalized wave function is obtained by ensuring that the integral of \(|\psi_n(x)|^2\) over the entire range is equal to 1.
\[ \int_{0}^{L} |\psi_n(x)|^2 \, dx = 1 \]
Solving this integral leads to the normalization constant:
\[A = \sqrt{\frac{2}{L}}\]
Derivation for Energy Eigenvalues:
The energy eigenvalues for the particle in a box are given by:
\[E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\]
These energy eigenvalues represent the quantized energy levels of the particle in the one-dimensional box.
Further Reading:
For a more in-depth understanding of the Particle in a One-Dimensional Box and related quantum mechanics concepts, consider exploring advanced textbooks and research papers in quantum physics.
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