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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Is it possible to give a perturbative potential to a free particle so that its energy levels...

The original system is a free particle, and there are no restrictions on its space. Assume that the perturbation potential is $\epsilon \frac{\mu\omega^2}{2} x^2$ so that $$ \hat{H}=\hat{H}_0+\epsilon …
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