Are All Solutions of the 1D Schrodinger Equation Energy Eigenstates?

[Zacarias Nason]

Just like it says, are all solutions of the 1D time independent Schrodinger equation, by default, energy eigenstates? I'm having a hard time imagining how solutions, with these conditions, that aren't energy eigenstates could exist if they have to satisfy the relation

$$E \psi(x)=\hat{H}\psi(x)$$

 

[Zacarias Nason]

And on that note, this statement would be true, correct?

$$\textbf{All} \ \text{energy eigenstates} \ \psi_n(x) \ \text{of the Hamiltonian} \ \hat{H} \ \text{associated with an energy eigenvalue} \ E_{n} \\ \ \text{are solutions of the time-independent Schrodinger equation} \ E\psi(x)=\hat{H}\psi(x).$$

(Coming at this from the angle of, "All sedans are vehicles, but not all vehicles are sedans"-all energy eigenstates are solutions of the 1D time-independent S.E., but potentially are not all solutions of the time-independent S.E. energy eigenstates?

 

[bhobba]

Of course it is by its very definition as the energy operator.

Thanks
Bill

 

[Zacarias Nason]

I honestly had a hard time understanding what "of course it is" was specifically in response to, but is it true to say that there are no solutions of the 1D time-independent S.E. that aren't energy eigenstates?

 

[bhobba]

Its because it follows directly from the foundational axioms of QM.

To every dynamical variable corresponds an operator whose eigenvalues are the possible outcomes of observing that variable.

For more detail see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Thanks
Bill

 

[Zacarias Nason]

I have a semantically/terminologically-based confusion right now, and putting that to rest or confirming what I already think is true when I am severely doubting my own reasoning sometimes requires somebody who knows what they are talking about just straightforwardly saying, "this statement is true, this is not", "yes" or "no". Thanks, though, I guess.

 

[dextercioby]

There's no such thing as "time-independent Schrödinger" equation. This wrongly used statement is unfortunately still propagated by some books.
The Schrödinger equation is either the original 1926 one involving potential energy and Laplace-ian, or the one in Dirac form from 1935:

$$\frac{d|\psi (t)\rangle}{dt} = \frac{1}{i\hbar} H |\psi (t)\rangle$$

$$H \psi = E\psi$$ is called spectral equation for the Hamiltonian operator in a (rigged) Hilbert space.

 

[A. Neumaier]

There's no such thing as "time-independent Schrödinger" equation. This wrongly used statement is unfortunately still propagated by some books.

Unfortunately, you propagate a both factually and historically wrong point of view.

The name "time-independent Schrödinger equation'' for the eigenvalue equation $H\psi=E\psi$ is accepted usage, used almost everywhere, and in particular in famous textbooks on quantum mechanics, e.g., in Messiah's textbook.
Entering "time-independent Schrödinger equation" (including the double quotation marks) gives 36.700 hits.

http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.1.157 by Kemble calls it ''the Schroedinger wave equstion'' (without any qualification)

Schrödinger himself derived the time-independent wave equation, eq. (5)-(6) in [Ann. Phys. 79 (1926), 361] (Erste Mitteilung) two journal volumes before discussing the time-dependent version, eq. (4'') in [Ann. Phys. 81 (1926), 109] (Vierte Mitteilung).

 

[stevendaryl]

And on that note, this statement would be true, correct?

$$\textbf{All} \ \text{energy eigenstates} \ \psi_n(x) \ \text{of the Hamiltonian} \ \hat{H} \ \text{associated with an energy eigenvalue} \ E_{n} \\ \ \text{are solutions of the time-independent Schrodinger equation} \ E\psi(x)=\hat{H}\psi(x).$$

(Coming at this from the angle of, "All sedans are vehicles, but not all vehicles are sedans"-all energy eigenstates are solutions of the 1D time-independent S.E., but potentially are not all solutions of the time-independent S.E. energy eigenstates?

I take "$|\psi\rangle$ is an energy eigenstate" to mean the same thing as "For some real number $E$: $H |\psi\rangle = E |\psi\rangle$.

 

[Jilang]

Isn't it possible to start with a wavefunction that is not an eigenstate and see how it evolves?

 

[A. Neumaier]

Isn't it possible to start with a wavefunction that is not an eigenstate and see how it evolves?

Of course. But then one needs the time-dependent SE. The time-independent SE describes only the initial conditions for those solutions that behave in a purely harmonic way - this singles out the eigenstates of the Hamiltonian.

 

[Jilang]

So Dexter's two equations are different? The first can be applied more generally; to states that are not eigenfunctions, but the second one is more specific?

 

[dextercioby]

Unfortunately, you propagate a both factually and historically wrong point of view.

The name "time-independent Schrödinger equation'' for the eigenvalue equation $H\psi=E\psi$ is accepted usage, used almost everywhere, and in particular in famous textbooks on quantum mechanics, e.g., in Messiah's textbook.
Entering "time-independent Schrödinger equation" (including the double quotation marks) gives 36.700 hits.

http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.1.157 by Kemble calls it ''the Schroedinger wave equation'' (without any qualification)

Schrödinger himself derived the time-independent wave equation, eq. (5)-(6) in [Ann. Phys. 79 (1926), 361] (Erste Mitteilung) two journal volumes before discussing the time-dependent version, eq. (4'') in [Ann. Phys. 81 (1926), 109] (Vierte Mitteilung).

Well, here you're venturing into my territory: first, let's settle the fact that a wave equation contains a time derivative, this is selbstverständlich and even "time- independent (Schrödinger or not) wave equation" is to be treated as an oxymoron from a linguistic standpoint (in the context of the so-called standing waves, the collocation wave equation is not used). The German word Wellengleichung appears on page 510 of Vol. 79 (Zweite Mitteilung) and will design his equation 18 on page 511. After getting rid of the 2nd order time derivative by means of a complex exponential, he gets two time-independent partial diff. equations (18' and 18''). Only on the 1st page of his Vierte Mitteilung does the word Wellengleichung appear again as you can see below.

[picture courtesy of http://gallica.bnf.fr/]

Since you brought up the review article by Kemble (1929), this is interesting, because he calls "wave equation" (the quotation marks used for emphasis are his) the standard D'Alembert equation of optics. When referring to Schrödinger's work, he quotes his "4th delivery" (from which the screenshot above has been taken), p. 109 and then 4th order order equation derived by Schrödinger (No. 4, p. 110) which through simplification comes back to equation 4'' of Schrödinger (op.cit., p. 112).

On a different note, as far back as 1928, Arnold Sommerfeld published his 1st German edition of "Wave Mechanics" (translated to English in 1930 and republished in 1936). In the English translation dated 1936, Sommerfeld calls his equation (11) which contains no time derivative the wave-equation but with the bottom-of-the-page note that, I quote, "Schrödinger himself originally wished to reserve the name <<wave-equation>> for one analogous to (5) [my note, this is D'Alembert's equation], but containing the time".

So this is for the old literature. Let's turn to the "new" one of the 1950s by Albert Messiah (book originally in French, English translation 1961). I quote:

A little lower we have that:

Then lower one has indeed:

So you see, you're right, indeed it's a pretty old convention to use this historical misnomer perpetuated by physicists, no wonder the Google search results back up your assertion. So, for me and those reading me, propagating this collocation detrimentally to the more mathematical one spectral equation for the Hamilton operator should be discouraged.

In the end an excerpt from FATHER DIRAC (1958, 4th Ed. of The Principles of Quantum Mechanics)

Screenshots courtesy of original copy-right holders.

 

[Zacarias Nason]

I have a genuinely super hard time believing that active MIT faculty are using terms outside of accepted usage both in their notes and in their lectures.

I take "$|\psi\rangle$ is an energy eigenstate" to mean the same thing as "For some real number $E$: $H |\psi\rangle = E |\psi\rangle$.

So, yes, then. There are no solutions to the time-independent [heresy!] 1D S.E. that are not energy eigenstates.

 

[dextercioby]

I didn't claim anything about "terms outside of accepted usage". Time-dependent or time-independent SE is undoubtedly old terminology, but as far back as 1958 Dirac clearly stated his point, see also the next screenshot

So you can go to the MIT people to tell them that they're teaching historically inaccurate and outdated terminology (do they teach out the wave-particle duality and relativistic mass, too??) propagated by books who only shadow what Hpsi = Epsi really stands for.

 

[bhobba]

I have a genuinely super hard time believing that active MIT faculty are using terms outside of accepted usage both in their notes and in their lectures.

I am with Dextercioby on this. MIT of course can counted on to use terminology that is generally accepted. Its just in QM there are a number of commonly used terms left over from the early days that are still in use - this is just another. Its nothing get worried about - just note it and move on.

Sometimes we have long threads on these sorts of things, many of which I have participated in. But really its nothing to get worked up about IMHO.

To the OP if you haven't seen it before glance at the following which looks at a number of generally accepted myths of QM:
http://arxiv.org/abs/quant-ph/0609163

Simply be aware of them, and sometimes papers and textbooks use them with gay abandon, which is usually only a problem in dealing with foundational issues. That's the one time we need to be careful.

Thanks
Bill

 

[Zacarias Nason]

Sometimes we have long threads on these sorts of things, many of which I have participated in. But really its nothing to get worked up about IMHO.

Simply be aware of them, and sometimes papers and textbooks use them with gay abandon, which is usually only a problem in dealing with foundational issues. That's the one time we need to be careful.

I'm now aware of them, it just needs to be understood I literally started studying QM six days ago and in the links supplied I haven't heard of several of the terms and have largely just become further confused and distanced from concrete, always-true and always-false statements in response to a specifically worded question-Everything said is likely true and good but for the purposes of something that won't necessitate me being mired in stuff for the moment way over my head, I still don't get it-I'm not saying I don't, I'm saying I probably won't be able to sink the time necessary to learn about the Heisenberg-Robertson uncertainty relation , gaussian spikes and PVM's adequately to get what you're trying to say before my homework is due in six days, lol.

I can either commit a phyrric victory and spend way too much time at the moment learning a detailed, correct and nuanced answer, or I can try to figure out "just enough to be dangerous". And thanks for everybody who addressed my question.

(And, to be clear, my OP question wasn't a homework question-I wasn't asked that-but it does obviously help advance my understanding so I can do said questions.)

 

[bhobba]

I literally started studying QM six days ago

Don't worry about these things for now - simply keep in the back of your mind some of the stuff has issues best sorted out later.

When I leaned QM I got caught up in that damnable Dirac Delta function. I did a long sojourn into esoteric areas to sort it out. I did - eventually - but I would have been better leaving it for later.

Thanks
Bill