Douglas: if you mean it will look either way as HY = EY, well, I meant more along the lines of “what does the hamiltonian look like for momentum space?”
HY = EY is not the Schrödinger equation—it is the energy eigenstate equation. The Schrödinger equation is i ℏ ∂t[Y] = H Y.
(EDITED TO NOTE: Markdownr’s sandbox renders the above correctly, but here it doesn’t come through right.)
As you said, that’s independent of basis. The Hamiltonian for a free spinless particle in momentum space is even more straightforward-looking than the hamiltonian in position space: k k / 2m + V(k). It doesn’t even contain any explicit derivatives!
Of course, the V(k) contains the Fourier transform of the potential.
All in all, I’m split between agreeing with Eliezer on the primacy of position, and saying ‘mu’.
What’s that Lincoln quote about ducks and calling things?
Point is, Schrodinger’s Equation contains within it an implication which leads to the energy eigenstate equation. Conflating the two is bad terminology, even if it’s common. I would not call the force balance equation from statics “Newton’s 2nd Law”—why should I do that in quantum mechanics, calling the Energy Eigenstate Equation “Schrodinger’s Equation”? My more recent textbook goes out of its way to separate the two as it was found that conflating them was impeding students’ understanding of quantum mechanics (though it does so in part by eliminating the term ‘Schrodinger Equation’ altogether).
That’s an entirely reasonable argument that it shouldn’t be called that.
But it is called that, and you have to be able to communicate with those who use it thus, or have it heard it this way, even while working to change the nomenclature.
All in all, I’m split between agreeing with Eliezer on the primacy of position, and saying ‘mu’.
Probably because the original post is actually a structureless rant. The only part that makes sense is
I accept the possibility that this whole blog post is merely stupid. After all, the question of whether the position basis or the momentum basis is “more fundamental” should never make any difference as to what we anticipate. If you ever find that your anticipations come out one way in the position basis, and a different way in the momentum basis, you are surely doing something wrong.
Douglas: if you mean it will look either way as HY = EY, well, I meant more along the lines of “what does the hamiltonian look like for momentum space?”
How does one actually write out the operator?
HY = EY is not the Schrödinger equation—it is the energy eigenstate equation. The Schrödinger equation is i ℏ ∂t[Y] = H Y.
(EDITED TO NOTE: Markdownr’s sandbox renders the above correctly, but here it doesn’t come through right.)
As you said, that’s independent of basis. The Hamiltonian for a free spinless particle in momentum space is even more straightforward-looking than the hamiltonian in position space: k k / 2m + V(k). It doesn’t even contain any explicit derivatives!
Of course, the V(k) contains the Fourier transform of the potential.
All in all, I’m split between agreeing with Eliezer on the primacy of position, and saying ‘mu’.
Which is often called the time-independent Schrödinger equation. The one with the d/dt is then called the time-dependent Schrödinger equation.
Typo: one instance of “dependent” (the first, if I’m reading Wikipedia correctly) needs to be “independent”.
Yep, fixing.
What’s that Lincoln quote about ducks and calling things?
Point is, Schrodinger’s Equation contains within it an implication which leads to the energy eigenstate equation. Conflating the two is bad terminology, even if it’s common. I would not call the force balance equation from statics “Newton’s 2nd Law”—why should I do that in quantum mechanics, calling the Energy Eigenstate Equation “Schrodinger’s Equation”? My more recent textbook goes out of its way to separate the two as it was found that conflating them was impeding students’ understanding of quantum mechanics (though it does so in part by eliminating the term ‘Schrodinger Equation’ altogether).
That’s an entirely reasonable argument that it shouldn’t be called that.
But it is called that, and you have to be able to communicate with those who use it thus, or have it heard it this way, even while working to change the nomenclature.
Probably because the original post is actually a structureless rant. The only part that makes sense is