A three-dimensional harmonic oscillator moves in the...

A three-dimensional harmonic oscillator moves in the potential $P(x, y, z)=\frac{1}{2} m \omega^{2}\left(x^{2}+y^{2}+z^{2}\right) .$ Separate the Schrödinger equation and show that the allowed energies are given by $E=\left(n+\frac{3}{2}\right) \hbar \omega .$

Key Concepts

Separation of Variables

This technique involves assuming that the solution to a partial differential equation (PDE) can be written as a product of functions, each depending on a single coordinate. In quantum mechanics, it is commonly used to reduce multidimensional problems into a set of one-dimensional problems, making complex systems more tractable. By separating the Schrödinger equation into components along each Cartesian axis, one can solve easier, standard one-dimensional equations.

Quantum Harmonic Oscillator

The quantum harmonic oscillator is a fundamental model in quantum mechanics that describes the motion of a particle in a quadratic potential well. Its significance lies in its solvability and its role as an approximation for more complex potentials near stable equilibrium points. The one-dimensional oscillator has discrete energy levels that increment in equal spacing, with a zero-point energy, and these properties can be extended to multiple dimensions.

Energy Quantization and Summation

In quantum mechanics, the energy levels of systems such as the harmonic oscillator are quantized, meaning that they can take on only specific discrete values. For systems involving separable dimensions, the total energy is the sum of the energies associated with each independent degree of freedom. In the case of a three-dimensional harmonic oscillator, each axis contributes its quantized energy and the overall energy is given by the sum of these contributions, incorporating the zero-point energies.

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