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My understanding from reading about quantum mechanics is that the state of a particle such as an electron can be kept in a superposition of energy states for an extended period of time, when it is not interacting with other particles, but that as soon as atoms interact with each other, their electrons will tend to an energy eigenstate.

But energy eigenstates are just one particular basis for representing quantum states of an electron. Why do particles tend to eigenstates of the hamiltonian, as opposed to elements of some random other basis?

user56834
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4 Answers4

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But energy eigenstates are just one particular basis for representing quantum states of an electron. Why do particles tend to eigenstates of the hamiltonian,

States don't "tend to" do anything other than evolve according to the well-known principles of Quantum Mechanics.

Setting aside measurement, states evolve in time according to the time-dependent Schrodinger equation.

When measured, states collapse to the eigenstate(s) of the measured observable, regardless of if the observable is energy, or position, or whatever.

The only energy eigenstate that you might reasonably say states "tend to" is the ground state. If a state has some small coupling to the outside world, considered as a heat bath, then at low enough temperature you would expect the state to be the ground state.

hft
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If a system is undergoing interference and information is copied out of that system that tends to suppress the interference: this process is called decoherence. For macroscopic objects this is usually a result of interaction with the environment and it selects a set of states that are robust under interaction with the environment that are highly localised in position and momentum and act approximately classical and such states aren't necessarily energy eigenstates: see Sections IV and V of Zurek's review Decoherence, einselection, and the quantum origins of the classical, see also

https://arxiv.org/abs/1111.2189

If interaction with the environment is relatively weak and is dominated by the self Hamiltonian this tends to prefer energy eigenstates.

The set of preferred states can be explained by decoherence, so collapse is unnecessary for explaining the states we see when we do observations. In addition you can only tell what states are preferred by working out the actual consequences of quantum equations of motion not by ad hoc, unclear and often unstated modifications of those equations of motion.

alanf
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Sometimes other basis states are interesting. Neutrino flavor eigenstates are not energy eigenstates. So netrinos oscillate, changing from one to the next.

FermiLab's Even Bananas series of video has a lot on neutrinos. In particular, How do neutrino oscillations work?

mmesser314
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The fundamental aspect to consider is what is the operator you are dealing with. Eigenvalues and Eigenstates are associated to specific operators; in the case of an energy operator, for example, such as the Hamiltonian describing the energy of a basic non-relativistic quantum system, the eigenvalues will be energy eigenvalues, and this can be proven by looking at the dimension of those eigenvalues $\lambda$ that satisfy the defining equation in operator form $H\psi=\lambda\psi$, where $H$ is the Hamiltonian of the system. Similarly, you have momentum eigenvalues for the momentum operator etc. Crucially, the eigenvalues and eigenstates are what completely describes the observables, i.e. the quantity than can/could be detected experimentally, so their determination is very informative. The determination of such operators that are physically meaningful is an important process in the construction of quantum theories. The construction of spin-operators, for example, are considerably less straightforward.

Note: I have not addressed the (considerably) trickier case of Lagrangian-based quantum systems but I believe this answer helps to shed light on OP's question.

Rimelius
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