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Illustration Energy quantization - quadratic potential energy function

Quantization of energy in the quadratic potential
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One-dimensional potential energy function \(W_{\text{pot}}(x)\) depends on the location \(x\). If a quantum mechanical particle is confined in the region between \(x_1\) and \(x_2\), its total energy \(W_n\) is always quantized. Then only energies \(W_0\), \(W_1\), \(W_2\) and so on are allowed, but no energy values in between. For each total energy of the particle there is the corresponding wave function \(\mathit{\Psi}_0\) (ground state), \(\mathit{\Psi}_1\), \(\mathit{\Psi}_2\) and so on (excited states).