Also, I guess I hadn't realized that this was supposed to have some connection with measurement - I thought the Schrödinger equation was just some equation that the wavefunction should always satisfy (though it's unclear to me why there should be some constant that we can call the "energy" for which the wavefunction is an eigenvector).
It's tough to find anything in quantum mechanics that isn't related to measurement somehow =) You don't have to think about measurement here, I guess, if you don't want to. It's just that the reason position and momentum are usually chosen as nice bases to work in, because they correspond *roughly* (but not exactly) to common types of measurements that are done. The Schrodinger equation in its abstract form is an operator equation, but the more commonly used "wave equation" that is also sometimes referred to as the Schrodinger equation (and was what Schrodinger was initially working with) is "the Schrodinger equation in the position basis". You can write it down in any basis, or you can just write it as an operator equation that is basis-independent. Each basis has a different set of eigenstates associated with it, that usually correspond to physical observables.
Regarding energy, in one sense it's no different from any other physical observable. The only thing that makes it special is that it is the conserved quantity associated with time-translation. Any system which has laws of physics that have no explicit reference to a specific time (like say, the big bang) must conserve energy, because of this relationship. This is true both in classical mechanics and in quantum mechanics, but one implication that has in quantum mechanics is that eigenstates of the energy operator (Hamiltonian) are time-independent. That's why energy eigenstates are special, they are sometimes referred to as "stationary" states since they do not change over time like most states do (actually, their phase can change, just not the magnitude which determines the underlying probability distribution).
Incidentally, when you say "for which the wavefunction is an eigenvector", that's not true of most wavefunctions. Most wavefunctions are not an eigenstate of energy... only stationary states are. However, stationary states are usually the first thing you want to find when you're studying a new quantum system, since it's very easy to build any more complicated time-dependent state out of a superposition of the simpler stationary states.
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
timeasmymeasure
no subject
Date: 2010-09-16 12:07 am (UTC)Also, I guess I hadn't realized that this was supposed to have some connection with measurement - I thought the Schrödinger equation was just some equation that the wavefunction should always satisfy (though it's unclear to me why there should be some constant that we can call the "energy" for which the wavefunction is an eigenvector).
It's tough to find anything in quantum mechanics that isn't related to measurement somehow =) You don't have to think about measurement here, I guess, if you don't want to. It's just that the reason position and momentum are usually chosen as nice bases to work in, because they correspond *roughly* (but not exactly) to common types of measurements that are done. The Schrodinger equation in its abstract form is an operator equation, but the more commonly used "wave equation" that is also sometimes referred to as the Schrodinger equation (and was what Schrodinger was initially working with) is "the Schrodinger equation in the position basis". You can write it down in any basis, or you can just write it as an operator equation that is basis-independent. Each basis has a different set of eigenstates associated with it, that usually correspond to physical observables.
Regarding energy, in one sense it's no different from any other physical observable. The only thing that makes it special is that it is the conserved quantity associated with time-translation. Any system which has laws of physics that have no explicit reference to a specific time (like say, the big bang) must conserve energy, because of this relationship. This is true both in classical mechanics and in quantum mechanics, but one implication that has in quantum mechanics is that eigenstates of the energy operator (Hamiltonian) are time-independent. That's why energy eigenstates are special, they are sometimes referred to as "stationary" states since they do not change over time like most states do (actually, their phase can change, just not the magnitude which determines the underlying probability distribution).
Incidentally, when you say "for which the wavefunction is an eigenvector", that's not true of most wavefunctions. Most wavefunctions are not an eigenstate of energy... only stationary states are. However, stationary states are usually the first thing you want to find when you're studying a new quantum system, since it's very easy to build any more complicated time-dependent state out of a superposition of the simpler stationary states.