Talk:Schrödinger equation

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Outline of Schrödinger equation for different levels

High school

  • A sketch of quantum mechanical phenomena such as the uncertainty principle, atomic energy levels and photons
  • Classical mechanics predicts the time dependence of properties like position, momentum and energy.
  • Quantum mechanics predicts the time evolution of probabilities.
  • Discuss discrete and continuous probabilities.
  • The double-slit experiment and wave-particle duality as an argument for desribing particles with a wave function

Heisenberg picture versus Schrödinger picture[edit]

Currently, the subsection Schrödinger equation#Matrix mechanics and the Schrödinger picture makes it appear wrongly that the equation

i\hbar{d\hat{A} \over dt} = [\hat{A},\hat{H}]= \hat{A}\hat{H} - \hat{H}\hat{A}

is universally true and consistent with the Schrödinger equation. On the contrary, that equation is only true in the Heisenberg picture (and then only when the partial derivative of A wrt time is zero) while the Schrödinger equation is only true in the Schrödinger picture. Although the two pictures give the same physical results, they are mathematically inconsistent with each other. (They use different bases for the Hilbert space as a function of time.) So any calculation should be done entirely in just one of the two pictures. This article should state at the outset that the Schrödinger equation is part of the Schrödinger picture, not the Heisenberg picture. JRSpriggs (talk) 09:34, 15 March 2012 (UTC)

I just moved this to Matrix mechanics without realizing this comment - sorry. -- F=q(E+v×B) ⇄ ici 09:21, 17 March 2012 (UTC)
Changes have been made here. I'm still in the process of cleaning up formatting and may re-locate it somewhere else in the article. That section (corrected) is better suited to the Matrix mech article than here.-- F=q(E+v×B) ⇄ ici 09:28, 17 March 2012 (UTC)

Atomic weight versus atomic number[edit]

Could you explain to me why atomic number Z is used in the equation in section Schrödinger equation#Two-electron atoms or ions in the kinetic energy term (and the formula for the reduced mass) rather than mass number A (or, more precisely, the mass of the nucleus as a whole including neutrons as well as protons divided by the mass of a proton)? Of course, Z is appropriate in the potential energy term. JRSpriggs (talk) 20:07, 3 April 2012 (UTC)

Zmp is the mass of the nucleus, mp is the mass of the proton. For 2-electron atoms mentioned there would be Z protons and... some neutrons. You're right. Lets replace with just M for the mass of the nucleus. F = q(E+v×B) ⇄ ∑ici 20:18, 3 April 2012 (UTC)
Fixed. Happy? Sorry for not thinking clearly again (2nd time it happened today, see here)... =( F = q(E+v×B) ⇄ ∑ici 20:22, 3 April 2012 (UTC)
Out of interest, isn't the reduced mass usually mu \mu\,\!, not the obscure m_{ep}\,\!? I have yet to see that in a source and arn't we supposed to be using the most common notation? Maschen (talk) 21:14, 3 April 2012 (UTC)
Don't remind me of that link - It’s been a bad day today. Does it matter? Greek \mu\,\! is overused, if it annoys you so much why don't you fix it?? F = q(E+v×B) ⇄ ∑ici 21:16, 3 April 2012 (UTC)
I will. Maschen (talk) 21:18, 3 April 2012 (UTC)

Introductory Preamble[edit]

I wanted to say something like:-

"Like Newton's second law Schrödinger's equation for a quantum system has not been proved or derived from more elementary principles. It is used because it works and has predicted a multitude of reactions time and time again - the only proof is in nature or natural." Blueawr (talk) 07:55, 1 September 2012 (UTC)

You could put that in somewhere, but it is the kind of statement that needs to be backed up with a good secondary source. RockMagnetist (talk) 16:53, 1 September 2012 (UTC)

Separate article on exact solutions?[edit]

I have proposed before (in passing) that we make an article on the exact solutions (and forgot, till now), such as Schrödinger equation (exact solutions), because;

written summary style for detailed articles on specific systems (step potential, delta potential, Harmonic oscillator (quantum), hydrogen atom etc.). Any objections? Maschen (talk) 09:59, 11 November 2012 (UTC)

Update: I'm drafting this in a pre-article (in namespace), it will not be created if people desire to re-merge in the future... Maschen (talk) 18:11, 1 December 2012 (UTC)

"Incorrect - why?"[edit]

Incorrect in that it doesn't detail the whole situation which would require something to be going faster when it is lower than when it is higher. Consider that a SR71 could be higher and be going faster (than I am), and you'll see why the statement is wrong. I think removal of the line is appropriate. --Izno (talk) 00:48, 6 December 2012 (UTC)

We're talking about the sentence "For example, a frictionless roller coaster has constant total energy; therefore it travels slower (low kinetic energy) when it is high off the ground (high gravitational potential energy) and vice versa." Right?
I read this as a statement about frictionless roller coasters, and I believe it is a correct statement about frictionless roller coasters. It seems to me that readers are unlikely to read this sentence and get the impression that it is true in any context besides frictionless roller coasters.
But maybe I'm wrong. So here is an idea:
  • "For example, a free, frictionless roller coaster has constant total energy; therefore this roller coaster will travel slower (low kinetic energy) when it is high off the ground (high gravitational potential energy) and vice versa." --Steve (talk) 16:20, 6 December 2012 (UTC)
I read it for what it was, and that it was trying to convey an example of conservation of energy. Your change makes it slightly better. But connecting a roller coaster to the law needs more; the assumptions are not laid out which would make the statement precisely true. On top of this, it still does not illustrate why it is the case that the roller coaster will travel faster in a different place. What is needed, quite frankly, is a link to the conservation of energy article. I still favor deletion of the line and would favor insertion of a link to the law of conservation of energy, which is quite oddly lacking for the first time that implications are discussed in the article. (Perhaps because it is used as a part of the assumptions section?) --Izno (talk) 17:11, 6 December 2012 (UTC)
In other words, it is not clear to me the connection. "It goes fast here and slow here" doesn't make it clear to a reader why that is the case, only that it is the case. --Izno (talk) 17:24, 6 December 2012 (UTC)
As the one who reverted, it should be clear that potential energy and kinetic energy interchange and the sum of these is constant assuming no dissipative effects (e.g. friction, heating). The worded equation is given there, no?. If the PE increases (for gravity, increase in altitude is one way) then the KE decreases (speed decreases by the square root), and vice versa. The aeroplane you linked to has nothing to do with roller coasters and frictionless systems so I have no clue why you point to that... Maschen (talk) 18:00, 6 December 2012 (UTC)
According to archive 3, the roller coaster example was added to analogize a familiar system (or at least one that is easy to understand) with a particle in a potential, i.e. the roller coaster tends to move in directions of decreasing potential, and there are stable/unstable equilibrium points. If you're finding this example confusing - maybe there is no harm in removing it... but it's not complicated... Maschen (talk) 18:07, 6 December 2012 (UTC)
I added the link conservation of energy in the first sentence of that section. Better? Maschen (talk) 20:23, 6 December 2012 (UTC)

Naturally, but this is only true for an object in motion with only one force (and conservative at that) acting upon it (or if there is a second force, it is a normal force and does not counteract completely the first force). I point to an airplane because it is a system which goes high and fast, which makes it a problematic counterexample to our example of a roller coaster. In effect, the comment of "high and slow" is so basic as to be useless, because it does not lay out the requisite assumptions.

If it is our desire to comment on the fact that systems act toward states with a lower potential, then the example certainly does not convey that.

The thought I had was to switch it with a pendulum, or a vibrating string. It's not uncommon to see either of the two in context of wave equations (in general). (I'll note that a pendulum is probably better for this example.) Even so, switching the type of system would not sufficiently fix the problem of the original statement. --Izno (talk) 20:38, 6 December 2012 (UTC)

Now I see what you are trying to say (except for when you said the example was wrong when it isn't, but we're past that now). Apologies... Perhaps it is too short to be fully understandable. The pendulum (or the like) sounds like a nice idea. Let's see what others think... Maschen (talk) 21:06, 6 December 2012 (UTC)
Actually (you'll like this), maybe the example could just be deleted since it only serves as an example of classical energy conservation in a specific case, suppose there is no loss of continuity. Up to others from this point on, I'll stay out of this... Maschen (talk) 21:16, 6 December 2012 (UTC)

Partial waves method[edit]

Something should be mentioned concerning the use of the equation to scattering problems by partial wave analysis.--188.26.22.131 (talk) 12:31, 2 August 2013 (UTC)

  • I don't agree. Of cause it is true that the Schroedinger equation is the basis of scattering theory. But this is true for basically any quantum mechanical problem. There is no point in listing all problems that build on this equation. Ciao. --Falktan (talk) 16:52, 26 August 2013 (UTC)

I'm not a physicist[edit]

Hey guys. I was curious about learning more about the cat in a box I heard and this page is really thick. You need a knowledge of quantum physics just to read the article. Any chance we can dumb it down for the lay person? — Preceding unsigned comment added by 216.106.18.70 (talk) 13:08, 12 August 2013 (UTC)

Maybe it needs something, somewhere, but this article is about physics. The "cat in a box business" was intended to make a serious point, but Schrödinger was a person who wrote ironically and sometimes sarcastically. His writings are frequently not to be taken exactly at face value. It may help to know that he disliked cats. It appears that his initial idea was to ridicule some of the early and emerging conclusions of the Copenhagen group by saying, in effect, "If you believe this bleep then you would have to accept the idea of a cat smooshed out in some state neither alive nor dead but in some sense both." Then, as the months and years rolled on, it became clear that electrons turned in both counterclockwise and clockwise rotation at the same time, and many other kinds of "superposition" could be demonstrated. Then, going back to the cat example, if you really did try to do the experiment, what would that mean? It turns out that the original quip left out a lot of detail that has been funneled together into the idea of decoherence. The original idea was that if nobody observes the geiger counter that is linked to a release mechcanism for the poison gas, then the geiger counter doesn't do anything, so the cat and everything in the box is suspended until a human being (delusions of grandeur here) opens the box and makes an observation. Who says that the cat cannot be aware of the geiger counter either clicking or not clicking? Who says that "observation" has to mean the report of stimuli received in human eyes or ears to brain tissue that processes the report somehow and concludes "the geiger counter has (not yet) clicked." Why won't a rolling movie camera in with the cat record the flickering needle of the geiger counter just as well? Why won't anything else that is done by being connected to the needle or the guts of the geiger counter serve just as well as an observation? And blah, blah, blah. You can't boil it down to a paragraph without making it into a dogmatic assertion that anybody with good sense should be suspicious of. The best you could do would be to say that Dr. Big explained/defined it that way ex cathedra. But it is, at heart, a big, complicated discussion. P0M (talk) 18:39, 12 August 2013 (UTC)
The "cat in a box" was supposed to illustrate the concept that two "impossible states" simultaneously occur (i.e.wavefunction) before one or the other does happen (as in wavefunction collapse).
Unfortunately, it seems one does need some exposure to the very basics of QM before the SE is introduced in most courses/textbooks I've seen, mainly in wave-particle duality which the article already does include, but given that it's a crucial aspect of QM (indeed a fundamental postulate) maybe we need to alter the background QM. For now, not sure how rewrite... M∧Ŝc2ħεИτlk 19:09, 12 August 2013 (UTC)
Note that there is an article on Schrödinger's cat. RockMagnetist (talk) 02:12, 13 August 2013 (UTC)

Article gives impression that the relativistic forms are also called the Schrödinger equation[edit]

I have not read through this thoroughly, but noticed the following that might confuse a reader:

  • The lead gives no indication that the domain of applicability of the Schrödinger equation is strictly the nonrelativistic domain. In fact, it says that it can be transformed into the Feynman path integral formulation, and since the latter is relativistic (I'm not even aware that it has any nonrelativistic form), I would go so far as to suggest that this is seriously misleading.
  • The article discusses both nonrelativistic and relativistic wave equations, giving the impression that the name "Schrödinger equation" encompasses them all. My understanding is that it refers strictly to the nonrelativistic equations given early on, and that the Dirac equation and the Klein–Gordon equation are not considered to be examples of the Schrödinger equation.
  • The subsections on time-dependent and time-independent equations would be better named for the respective forms of the Hamiltonian, not the resultant form of the equation, e.g. General Hamiltonian and Time-independent Hamiltonian. They could also be written to emphasise that with a time-independent Hamiltonian (which in itself is generally inherently an approximation) the equation can take another form

My involvement, time and expertise in this sort of article is not sufficient to make such edits, but I hope any who are suitably interested would like to use these pointers as inputs to their edits. — Quondum 06:30, 23 September 2013 (UTC)

The Schro eqn is a fundamental postulate of QM, and the article states the general equation can be used in relativistic and non-relativistic context (yes, it can, all that's needed is the correct Hamiltonian), if this were not the case then how can non-relativistic QM and RQM be consistent with the postulates of QM?
Feynman path integrals are equivalent to the Schrodinger equation, and can be relativistic or non-relativistic (there is a connection between non-relativistic QM and classical mechanics... the correspondence principle). There is plenty literature on Feynman path integrals, but I don't have any to hand right now. For a WP pointer, this section in the path integral formulation article uses the classical kinetic energy expression.
A "general Hamiltonian" is time-dependent in general anyway, and is more meaningful than "general" and provides contrast with "time-independent".
M∧Ŝc2ħεИτlk 05:59, 24 September 2013 (UTC)
The article says "The general form of the Schrödinger equation is still applicable, but the Hamiltonian operators are much less obvious, and more complicated." Such an equation strikes me as highly contrived; for example, the Hamiltonian needed to obtain the Klein–Gordon equation would be really strange, and the Dirac equation cannot be considered to be a Schrödinger equation, as the Dirac equation is in terms of matrices, and the Schrödinger equation in terms of real values. Is this really in sources?
I'll accept what you say about Feynman path integrals. — Quondum 01:23, 25 September 2013 (UTC)
Yes, the Dirac equation can be considered a Schro equation, Penrose says this in the chapter on the Dirac equation (page 621 in my copy "the Dirac equation can be written in the form of a Schro equation ... Of course the singling out of the time derivative is not relativistically invariant but the entire Dirac equation is"). Particle physics books (many are listed in the RQM article references, the one I have to hand now is Particle physics by B.R. Martin and G. Shaw) usually say something like "Dirac proposed a Hamiltonian of the form:
\hat{H}=\boldsymbol{\alpha}\cdot \hat{\mathbf{p}} + \beta mc
where a = α1, α2, α3 and β are to be found subject to the constraints..."
While non-relativistic quantum Hamiltonians are just functions of position, momentum, and time (as in classical Hamiltonian mechanics), the relativistic quantum Hamiltonians are also functions of position, momentum, time, and spin matrices. The Schro equation does not restrict the wavefunction to be a scalar, it could be anything that can describe quantum mechanics.
You have a point about the KG Hamiltonian, most people don't derive it from the Hamiltonian since it can be derived directly from the energy-momentum relation or viewed as the square of the Dirac equation (you knew that), but I'm sure it's possible one way or another. I'll look for sources.
M∧Ŝc2ħεИτlk 07:37, 25 September 2013 (UTC)
By the way, a good faith edit was made to the RQM section of this article saying QM is formulated to be consistent with SR. As YohanN7 confirmed a while back (see talk:relativistic quantum mechanics#Suggestions): RQM = SR + QM, i.e. special relativity and quantum mechanics applied together. The statement "Quantum mechanics formulated to be consistent with special relativity" may have the implication that QM is modified in some way (or may not...). Quantum mechanics doesn't actually change, all pictures of QM are already applicable with SR, but the outcome is that new predictions and mathematical objects appear, which did not from QM alone. M∧Ŝc2ħεИτlk 07:56, 25 September 2013 (UTC)
I suppose the Schrödinger equation evolved rather than being replaced. This is in accordance with the physicist's natural approach of "if it works, use it", in contrast with the mathematician's more rigorous approach. Given that the Schrödinger equation can be adapted to both Galilean and special relativity (and no doubt anything else), that the wavefunction and the underlying abstract algebra is regarded as being free in type, and that the Hamiltonian is unconstrained (it presumably becomes only loosely representative of energy), the Schrödinger equation is essentially not falsifiable, which is not a pretty state of affairs. It asserts that there exists some wavefunction, which when subject to some operator (which is expressed as the difference between the "Hamiltonian" and the partial derivative with respect to time multiplied by a constant), yields zero. More strictly interpreted, it says that the wavefunction is the solution of a homogenous differential equation, or equivalently that it obeys the superposition principle - and nothing else whatsoever. Beyond this, anything to be added is freely chosen in the choice of abstract algebra and Hamiltonian.
On my edit, I have reverted it since as you suggest, there should not be a suggestion of reformulation of QM, only that its form is constrained to be relativistically invariant. I am not particularly comfortable with "applied together", due to what I perceive as a semantic difference between "applied together" and "simultaneously apply". The former is an active form, suggesting a process in which one takes QM, and then applies SR, getting something new. The latter is a passive form, indicating that they both are part of the framework, i.e. they act as constraints. Would this change make more sense? — Quondum 09:53, 25 September 2013 (UTC)
The second change, for the RQM section, seems fine with me.
Remember the Schrodinger equation is a postulate of quantum mechanics, assumed to be true, so it could be falsified (just like the postulates of special/general relativity) if there was some experimental phenomenon which the SE cannot describe (the Hamiltonian for the phenomenon is impossible to obtain or just pathologically unphysical).
I'm not sure what the issue is with the wavefunction, that in itself is another, separate, postulate. The SE is the evolution equation for it. It was constructed to be linear so that the superposition of states applies. What about it? M∧Ŝc2ħεИτlk 10:18, 25 September 2013 (UTC)
The article states:
This is the equation of motion for the quantum state. In the most general form, it is written:
i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi.
With no constraint on the Hamiltonian other than it be a linear operator field over time and space, or on the underlying algebra other than that it be compatible with complex scalar multiplication, the general Schrödinger equation can be rewritten as
\hat P \Psi = 0
for an arbitrary linear operator \hat P.
This is why I said what I said before: with no further constraints, one can deduce linearity, but nothing else. The only way of falsifying this equation is if linearity does not apply, and I don't think that linearity is in contention. If there are some constraints on \hat P (equivalently the Hamiltonian), for example that it obeys some conditions (e.g. locality, which would manifest as the operator being expressible in terms of partial derivatives with respect to the space and time coordinates), the equation would assert more than superposition. For the Galilean and special relativity cases, further constraints would apply. Even so, it is still saying very little until one adds yet further constraints, such as conservation of energy and momentum. So, I hope you can see that pretty much everything that can be "deduced" (including that it is an equation of continuity/evolution) must be built into the form of (or constraints on) the Hamiltonian. — Quondum 12:12, 25 September 2013 (UTC)
Yes, I do know that constraints are needed on the Hamiltonian in the context to construct it using physical principles, but I'm still not 100% sure what you're trying to get at improving the article...
OK... This article already gives the construction for the non-relativistic Hamiltonian, for one and n particles in any potential, we can agree on that. Are you saying to discuss the Hamiltonian (with constraints) for other situations in this article, like for the Pauli equation, and relativistic wave equations in general? We could always rewrite the article in some places (maybe have a section on this after some reorganizing), linking to the remaining details in the Hamiltonian operator article.
I apologize for misunderstanding... M∧Ŝc2ħεИτlk 19:33, 25 September 2013 (UTC)
I need to review the article in more detail to make specific suggestions, which I will be able to do more sensibly now that you have disabused me of some of my preconceptions. I should then be able to make more specific suggestions. — Quondum 21:43, 25 September 2013 (UTC)

Split off the non-relativistic content to another article(s)?[edit]

In response to the above thread...

I proposed before (more than once now) to transfer the exact solutions of the Schro eqn to a separate article Schrödinger equation (exact solutions). Note also the main article quantum mechanics has overlap on exact solutions to the non-relativistic Schro equation. Would cutting out the "exact solution" sections in this article and the QM article be useful?

Or instead, we could just transfer all of the non-relativistic formalism to another article: Schrödinger equation (non-relativistic) (by all means correct me on the dashes, this is just a suggestion...). This may have been proposed by someone before also...

Maybe both? In Schrödinger equation (non-relativistic), we have a discussion of the general ideas of the non-relativistic case, leading onto the mathematical details (such as those in the "properties" section of this article), without the exact solutions, then linking to the Schrödinger equation (non-relativistic) article.

Any thoughts? M∧Ŝc2ħεИτlk 20:36, 25 September 2013 (UTC)

Update: see also this section. M∧Ŝc2ħεИτlk 12:51, 27 September 2013 (UTC)
I just realized there is the article List of quantum-mechanical systems with analytical solutions. So there is no need for a new article. M∧Ŝc2ħεИτlk 09:16, 27 January 2014 (UTC)

Fails to Specify Variables[edit]

In a lot of the equations it doesn't specify the meanings of all the variables. This isn't "quantum physics experts we already know all this" wiki. Every variable should be specified as to what it means.35.8.84.29 (talk) 19:52, 22 October 2013 (UTC)

Specifically the backwards 6s.35.8.84.29 (talk) 19:59, 22 October 2013 (UTC)

For almost every equation the variables are specified, and there are links to the related articles like De Broglie relations and partial differential equations. Nevertheless, I agree there are some notational gaps (such as partial derivatives - reverse "6"), and will try to fill them in. M∧Ŝc2ħεИτlk 15:27, 23 October 2013 (UTC)
Done. Please point out any specific equations which lack definitions of symbols. Thanks, M∧Ŝc2ħεИτlk 15:43, 23 October 2013 (UTC)

Axiomatic construction using time-evolution operator[edit]

I guess the article is missing the key concept of axiomatic derivation of the Schrodinger's equation using the time evolution operator. This is the most natural and modern approach to understanding the equation (although, most likely, not originally proposed by Schrodinger). The details can be found, for example, in "Modern Quantum Mechanics" by J. J. Sakurai or this lecture by Prof. Susskind. - Subh83 (talk | contribs) 22:07, 21 February 2014 (UTC)