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Quantum mechanics
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=== Harmonic oscillator === {{Main|Quantum harmonic oscillator}} [[File:QuantumHarmonicOscillatorAnimation.gif|thumb|upright=1.35|right|Some trajectories of a [[harmonic oscillator]] (i.e. a ball attached to a [[Hooke's law|spring]]) in [[classical mechanics]] (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [[wave]] (called the wave function), with the [[real part]] shown in blue and the [[imaginary part]] shown in red. Some of the trajectories (such as C, D, E, and F) are [[standing wave]]s (or "[[stationary state]]s"). Each standing-wave frequency is proportional to a possible [[energy level]] of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]] As in the classical case, the potential for the quantum harmonic oscillator is given by<ref name="Zwiebach2022" />{{rp|234}} <math display=block>V(x)=\frac{1}{2}m\omega^2x^2.</math> This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The [[eigenstate]]s are given by <math display=block> \psi_n(x) = \sqrt{\frac{1}{2^n\, n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad </math> <math display=block>n = 0,1,2,\ldots. </math> where ''H<sub>n</sub>'' are the [[Hermite polynomials]] <math display=block>H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right),</math> and the corresponding energy levels are <math display=block> E_n = \hbar \omega \left(n + {1\over 2}\right).</math> This is another example illustrating the discretization of energy for [[bound state]]s.
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