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Energy eigenstates provide a convenient basis for solving quantum mechanics problems, but they are by no means the only allowable states. Yet it seems to me that particles/systems are assumed to be in energy eigenstates "in nature".

Some examples of what I mean:

  • Solving the Schrodinger equation for the Hydrogen atom gives the standard $|n,l,m \rangle$ basis of energy/angular momentum eigenstates. But we speak of "filling up" these orbitals with electrons, or transitions between energy levels. Why should we expect to find the electrons only in such energy eigenstates, as opposed to say, some arbitrary superposition?
  • In quantum statistical mechanics we have Bose-Einstein and Fermi Dirac distributions that give the number particles in a state of energy $\epsilon$, but why must a particle be found in a state of definite energy to begin with?
yuval
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4 Answers4

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If we take a system, and let it evolve for some indefinite amount of time, it will be in an incoherent mixture of energy eigenstates. Many systems we encounter in nature have been sitting for some time, and not interacting with the environment (much). These can be considered to be in energy eigenstates.

For example, suppose we consider an atom in a gas. Let's assume that the last interaction it had with the environment was a collision with another atom, which put it in some state which we can consider as a superposition of energy eigenstates. Now, let's look at it. Even if we know everything about the collision except how long ago it collided, we can't determine the phase of this superposition, so we might as well think about it as being in a probabilistic mixture of energy eigenstates.

Peter Shor
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Is there a more sophisticated answer than "This is done because isolated low-temperature systems tend to give up energy more often than they receive it, and thus, gravitate toward their ground states. Thus, even if they start in a mixed energy state, the system will radiate away energy until it reaches its ground state, which is obviously an energy eignenstate."?

Zo the Relativist
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  1. Let me start with a "sociological" remark: it seems to me that many people do believe (erroneously) that a system can only be in an eigenstate of energy. I guess this may have some historical roots - IMHO, some of the great founders of quantum mechanics did emphasize "quantum jumps" more than they should have.

  2. I would say "a convenient basis for solving quantum mechanics problems" contains not just "energy eigenstates", those states are typically also eigenstates of the components of total momentum of the system.

  3. innisfree mentions in his/her comment that the Hamiltonian is "the generator of time translations". So maybe the reason that eigenstates of total energy and momentum form a convenient basis is the homogeneity of space and time. A similar argument is valid for angular momentum as well, but different components of angular momentum do not commute.

akhmeteli
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Heisenberg discovers Quantum mechanics (Matrix mechanics), by considering transitions between different energy levels of. In fact, the position $X(t)$ is no more a simple quantity, but has to be written $X_{ij}(t)$, because it is a transition between 2 energy levels $i$ and $j$, so $X(t)$ is a matrix (in fact, it is an infinite matrix, so it is called an operator). For instance, for the harmonic oscillator, we may write :

$$X(t) \sim ae^{-i\omega t} + a^+e^{i\omega t}$$ with $[a,a^+]=1$

Here $a$ and $a^+$ are operators which make the transition between energy levels of index $n$ and $n+1$

So the origin of Quantum Mechanics is clearly based on energy levels, and the position operator $X(t)$ is clearly made of transitions between energy levels. So it is not surprising that the energy basis is more fundamental in Quantum mechanics.

Trimok
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