Transcription of The Quantum Harmonic Oscillator
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The Quantum Harmonic OscillatorStephen WebbThe Importance of the Harmonic OscillatorThe Quantum Harmonic Oscillator holds a unique importance in Quantum mechanics, as it is both oneof the few problems that can really be solved in closed form, and is a very generally useful solution, both inapproximations and in exact solutions of various Harmonic Oscillator is characterized by the Hamiltonian: H=P22m 12m 2X2 This Hamiltonian appears in various applications, and in fact the approximation of the Harmonic oscillatoris valid near the minimum of any potential function. Expanded around a minimum point x*, any potentialcan then be Taylor expanded as:V x =V0 x x* V x x=x* 12 x x* 2 2V x2 x=x* In this case, V0 is an arbitrary constant that can be taken to zero, and the second term is zero since thederivative is zero at critical points.
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one
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