Answer to Question #106434 in Quantum Mechanics for Nikita Koladiya

Question #106434
Consider a proton as a bound oscillator with a natural frequency of Calculate the energy of its ground and first excited states.
1
Expert's answer
2020-03-25T10:59:35-0400

The energy levels of bound harmonic oscillator are

(1) "E_n=h\\nu\\cdot (n+\\frac{1}{2}), n=0,1,2,..."

The ground state must correspond to a stable particle - the proton and its rest energy. Proton has rest energy "\\epsilon=938 MeV" . This value corresponds to the oscillator frequency, which can be found from the equality of the ground state energy and the photon quantum "\\frac{1}{2}h\\nu=\\epsilon; \\\\\\nu=\\frac{2\\epsilon}{h}=\\frac{2\\cdot 9,38 10^8eV}{4.14\\cdot10^{\u221215}eVs}=4.53\\cdot 10^{23}Hz"

The value of the Planck constant bar expressed in various quantities can be found in [1].

The first excited states have energy "E_n=\\frac{3}{2}\\cdot h\\nu=3\\epsilon=2.81\\cdot 10^3 MeV"

Answer: The energy of ground and first excited states of proton are "938 MeV" and "2.81\\cdot 10^3 MeV" respectively.

[1] https://en.wikipedia.org/wiki/Planck_constant


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