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In many books, they refer summery to a conserved physical quantity, this is know as Noether's theorem in classical Mechanics. In quantum mechanics, the degeneracy of energy level is related the hamiltonian symmetry. The degeneracy is the fact that a unique eigenvalue has two or more eigenstates which are linearly independent. Physically, I can't grasp the fact that different states of the a system has the same energy. How could this happen. Is there a concrete example to accept this evidence?

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    $\begingroup$ An airplane headed due east and an airplane headed due west can be in different states with the same energy. $\endgroup$
    – WillO
    Apr 13, 2020 at 21:14
  • $\begingroup$ That's make sense, For a fixed value of momentum, each direction define a state with the same energy. $\endgroup$ Apr 13, 2020 at 21:17

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Here's an example that works for both classical and quantum mechanics. Imagine you have a particle of mass $m$ in free space, with momentum $\vec p =p\hat{x}$. Then the energy is $\frac{p^2}{2m}$. Say the particle instead has $\vec p = p\hat y$. Then the state of the particle is different, but the energy is the same. The energy operator is highly degenerate, since any momentum of the form $\vec{p}=p\cos(\theta)\hat x+p\sin(\theta)\hat y$ has the same energy $\frac{p^2}{2m}$. This, of course, comes from the rotation symmetry of the problem.

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  • $\begingroup$ I grasped it, and now I can imagine why a free particle have an infinite degeneracy of each momentum. Rather than bound states have a finite degeneracy. Thank you a lot. $\endgroup$ Apr 13, 2020 at 21:04
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    $\begingroup$ A bound classical system can also have infinite degeneracy. For example, the energy of an elliptical planetary orbit depends only on the semi-major axis and not on the eccentricity or the orientation in space. $\endgroup$
    – G. Smith
    Apr 13, 2020 at 22:11
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Some energy eigenstates of the hydrogen atom are degenerate. The energy $E_n$ depends only on the quantum number $n$ but many states can share this $n$. For instance, with $n=2$ there are $3$ states with $\ell=1$ plus another state with $\ell=0$ all having the same energy $-13.6/4$ eV.

Of course in classical mechanics the evolution of a system will take place on a surface of constant $H$ (when $L$ does not depend on time) so all points of that surface are examples of degenerate states.

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  • $\begingroup$ In classical mechanics, if we consider hydrogen atom, it is hard to imagine a finite degeneracy for bound sates because the energy is a continuous quantity. In contrast, quantum mechanics, it is clear about that. $\endgroup$ Apr 13, 2020 at 21:28

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