Perhaps “multiverse” would have been a better word.
Imagine an electron in an eigenstate. It’s moving around the atom, but the entire waveform is stationary. Similarly, if the entire universe is in an eigenstate, the universe would still be moving, but the probability for a given universe would stay constant. Put another way, as time passes in this universe, a parallel universe just like this one was moments ago moves into its place. We have no way of proving that this doesn’t happen. In fact, to some extent, it must. The immediate past is almost exactly like the present, so a universe that looks just like it must have similar amplitude to this one now.
Do you speak about calculating eigenfunctions of the Hamiltonian, then? If so, it is not such a big problem numerically. In one spatial dimension it can be done in a straightforward fashion: select an energy E, chose two distinct doubles of values of ψ(x) and dψ(x)/dx at some point x0, numerically solve the Schrödinger equation for both values at x0, finding ψ1(x) and ψ2(x), look whether you can find a linear combination ψ = a ψ1 + b ψ2 so that ψ(x) = 0 at both boundaries, move over all values of E to find those when this is possible, those are the eigenvalues.
In more spatial dimension you can do the same if you can separate the variables. If not, you can still do something. For example, if you want to find the ground state, first make sure that the Hamiltonian is positive (if not, add a constant). Then begin with arbitrary wave function (which satisfies the boundary conditions, but isn’t stationary) and let it evolve in imaginary time, i.e. solve the Schrödinger equation without i at the time derivative. All components will be supressed exponentially, but the lowest energy component will vanish most slowly (or remain constant if you managed to choose the additive constant so that the lowest energy is zero). Effectively you get the ground state wave function.
By the way, I find the “Timeless Physics” label somewhat confusing here. As I understand it, it is a way how to look at physics, not giving time a special role. Whatever one thinks about ontological status of time shouldn’t influence whether one is interested in stationary solutions or not.
Do you speak about calculating eigenfunctions of the Hamiltonian, then?
That would only tell you what values are possible for E, not what to set ψ(0) and ψ′(0) to, wouldn’t it? That said, I doubt finding those would be much harder.
As I understand it, it is a way how to look at physics, not giving time a special role.
Not exactly. It’s a different theory. If you believe time exists, the past and future wouldn’t interfere with each other. If you believe it doesn’t, there’s only one “time” and everything interferes. To my knowledge, it’s not different enough to actually test, like MWI vs. Copenhagen, but it’s different.
That would only tell you what values are possible for E, not what to set ψ(0) and ψ′(0) to,
They are already calculated in the process—they are a ψ1(0) + b ψ2(0), or with primes respectively.
Not exactly. It’s a different theory. If you believe time exists, the past and future wouldn’t interfere with each other. If you believe it doesn’t, there’s only one “time” and everything interferes.
I don’t understand what interference of past and future means.
They are already calculated in the process—they are a ψ1(0) + b ψ2(0), or with primes respectively.
Wouldn’t that just make it so it’s zero at that point, not that it doesn’t increase without limit as you get further?
I don’t understand what interference of past and future means.
According to normal MWI, there’s a universe in the present that’s just like this one was in the immediate past. Each of them have their own amplitude. If you wanted to calculate the probability of a universe that looks exactly like that, you’d take the probabilities of those universes individually and add them together. With timeless physics, there’s only one universe that looks like that. If you were to add two waveforms, perhaps one representing then and one now, you’d add the amplitudes and then square the magnitude of that.
I try to make a ψ1 + b ψ2 zero at boundaries, not at origin. If the boundaries lie in +- infinity, I have to approximate it by some finite number (when I was doing that last time, I used infinity = 8, it worked well with the respective problem). If this is what you have asked about.
According to normal MWI, there’s a universe in the present that’s just like this one was in the immediate past. Each of them have their own amplitude. If you wanted to calculate the probability of a universe that looks exactly like that, you’d take the probabilities of those universes individually and add them together.
I’m sorry, but I can’t parse the demonstrative pronouns. Please denote the universes by symbols.
I’m not certain what you mean by time independent quantum mechanics in this case. Do you mean identifying energy eigenstates and their eigenvalues?
I mean Timeless Physics. I accidentally called it “time independent” because it’s governed by the time-independent Schroedinger equation.
It’s basically equivalent to assuming that the entire universe is a standing wave.
But the universe isn’t in an energy eigenstate. Things have been known to happen, from time to time, which they wouldn’t if it were.
Perhaps “multiverse” would have been a better word.
Imagine an electron in an eigenstate. It’s moving around the atom, but the entire waveform is stationary. Similarly, if the entire universe is in an eigenstate, the universe would still be moving, but the probability for a given universe would stay constant. Put another way, as time passes in this universe, a parallel universe just like this one was moments ago moves into its place. We have no way of proving that this doesn’t happen. In fact, to some extent, it must. The immediate past is almost exactly like the present, so a universe that looks just like it must have similar amplitude to this one now.
Do you speak about calculating eigenfunctions of the Hamiltonian, then? If so, it is not such a big problem numerically. In one spatial dimension it can be done in a straightforward fashion: select an energy E, chose two distinct doubles of values of ψ(x) and dψ(x)/dx at some point x0, numerically solve the Schrödinger equation for both values at x0, finding ψ1(x) and ψ2(x), look whether you can find a linear combination ψ = a ψ1 + b ψ2 so that ψ(x) = 0 at both boundaries, move over all values of E to find those when this is possible, those are the eigenvalues.
In more spatial dimension you can do the same if you can separate the variables. If not, you can still do something. For example, if you want to find the ground state, first make sure that the Hamiltonian is positive (if not, add a constant). Then begin with arbitrary wave function (which satisfies the boundary conditions, but isn’t stationary) and let it evolve in imaginary time, i.e. solve the Schrödinger equation without i at the time derivative. All components will be supressed exponentially, but the lowest energy component will vanish most slowly (or remain constant if you managed to choose the additive constant so that the lowest energy is zero). Effectively you get the ground state wave function.
By the way, I find the “Timeless Physics” label somewhat confusing here. As I understand it, it is a way how to look at physics, not giving time a special role. Whatever one thinks about ontological status of time shouldn’t influence whether one is interested in stationary solutions or not.
That would only tell you what values are possible for E, not what to set ψ(0) and ψ′(0) to, wouldn’t it? That said, I doubt finding those would be much harder.
Not exactly. It’s a different theory. If you believe time exists, the past and future wouldn’t interfere with each other. If you believe it doesn’t, there’s only one “time” and everything interferes. To my knowledge, it’s not different enough to actually test, like MWI vs. Copenhagen, but it’s different.
They are already calculated in the process—they are a ψ1(0) + b ψ2(0), or with primes respectively.
I don’t understand what interference of past and future means.
Wouldn’t that just make it so it’s zero at that point, not that it doesn’t increase without limit as you get further?
According to normal MWI, there’s a universe in the present that’s just like this one was in the immediate past. Each of them have their own amplitude. If you wanted to calculate the probability of a universe that looks exactly like that, you’d take the probabilities of those universes individually and add them together. With timeless physics, there’s only one universe that looks like that. If you were to add two waveforms, perhaps one representing then and one now, you’d add the amplitudes and then square the magnitude of that.
I try to make a ψ1 + b ψ2 zero at boundaries, not at origin. If the boundaries lie in +- infinity, I have to approximate it by some finite number (when I was doing that last time, I used infinity = 8, it worked well with the respective problem). If this is what you have asked about.
I’m sorry, but I can’t parse the demonstrative pronouns. Please denote the universes by symbols.