Answer to Question #106434 in Quantum Mechanics for Nikita Koladiya

Question

Consider a proton as a bound oscillator with a natural frequency of Calculate the energy of its ground and first excited states.

Answer 1

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 $\frac{1}{2}h\nu=\epsilon; \\\nu=\frac{2\epsilon}{h}=\frac{2\cdot 9,38 10^8eV}{4.14\cdot10^{−15}eVs}=4.53\cdot 10^{23}Hz$

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