You Should Love (or at least respect) the Schrödinger Equation

Over at the twitter dot com website, there has been a briefly-trending topic #fav7films, discussing your favorite seven films. Part of the purpose of being on twitter is to one-up the competition, so I instead listed my #fav7equations. Slightly cleaned up, the equations I chose as my seven favorites are:

  1. {\bf F} = m{\bf a}
  2. \partial L/\partial {\bf x} = \partial_t ({\partial L}/{\partial {\dot {\bf x}}})
  3. {\mathrm d}*F = J
  4. S = k \log W
  5. ds^2 = -{\mathrm d}t^2 + {\mathrm d}{\bf x}^2
  6. G_{ab} = 8\pi G T_{ab}
  7. \hat{H}|\psi\rangle = i\partial_t |\psi\rangle

In order: Newton’s Second Law of motion, the Euler-Lagrange equation, Maxwell’s equations in terms of differential forms, Boltzmann’s definition of entropy, the metric for Minkowski spacetime (special relativity), Einstein’s equation for spacetime curvature (general relativity), and the Schrödinger equation of quantum mechanics. Feel free to Google them for more info, even if equations aren’t your thing. They represent a series of extraordinary insights in the development of physics, from the 1600’s to the present day.

Of course people chimed in with their own favorites, which is all in the spirit of the thing. But one misconception came up that is probably worth correcting: people don’t appreciate how important and all-encompassing the Schrödinger equation is.

I blame society. Or, more accurately, I blame how we teach quantum mechanics. Not that the standard treatment of the Schrödinger equation is fundamentally wrong (as other aspects of how we teach QM are), but that it’s incomplete. And sometimes people get brief introductions to things like the Dirac equation or the Klein-Gordon equation, and come away with the impression that they are somehow relativistic replacements for the Schrödinger equation, which they certainly are not. Dirac et al. may have originally wondered whether they were, but these days we certainly know better.

As I remarked in my post about emergent space, we human beings tend to do quantum mechanics by starting with some classical model, and then “quantizing” it. Nature doesn’t work that way, but we’re not as smart as Nature is. By a “classical model” we mean something that obeys the basic Newtonian paradigm: there is some kind of generalized “position” variable, and also a corresponding “momentum” variable (how fast the position variable is changing), which together obey some deterministic equations of motion that can be solved once we are given initial data. Those equations can be derived from a function called the Hamiltonian, which is basically the energy of the system as a function of positions and momenta; the results are Hamilton’s equations, which are essentially a slick version of Newton’s original {\bf F} = m{\bf a}.

There are various ways of taking such a setup and “quantizing” it, but one way is to take the position variable and consider all possible (normalized, complex-valued) functions of that variable. So instead of, for example, a single position coordinate x and its momentum p, quantum mechanics deals with wave functions ψ(x). That’s the thing that you square to get the probability of observing the system to be at the position x. (We can also transform the wave function to “momentum space,” and calculate the probabilities of observing the system to be at momentum p.) Just as positions and momenta obey Hamilton’s equations, the wave function obeys the Schrödinger equation,

\hat{H}|\psi\rangle = i\partial_t |\psi\rangle.

Indeed, the \hat{H} that appears in the Schrödinger equation is just the quantum version of the Hamiltonian.

The problem is that, when we are first taught about the Schrödinger equation, it is usually in the context of a specific, very simple model: a single non-relativistic particle moving in a potential. In other words, we choose a particular kind of wave function, and a particular Hamiltonian. The corresponding version of the Schrödinger equation is

\displaystyle{\left[-\frac{1}{\mu^2}\frac{\partial^2}{\partial x^2} + V(x)\right]|\psi\rangle = i\partial_t |\psi\rangle}.

If you don’t dig much deeper into the essence of quantum mechanics, you could come away with the impression that this is “the” Schrödinger equation, rather than just “the non-relativistic Schrödinger equation for a single particle.” Which would be a shame.

What happens if we go beyond the world of non-relativistic quantum mechanics? Is the poor little Schrödinger equation still up to the task? Sure! All you need is the right set of wave functions and the right Hamiltonian. Every quantum system obeys a version of the Schrödinger equation; it’s completely general. In particular, there’s no problem talking about relativistic systems or field theories — just don’t use the non-relativistic version of the equation, obviously.

What about the Klein-Gordon and Dirac equations? These were, indeed, originally developed as “relativistic versions of the non-relativistic Schrödinger equation,” but that’s not what they ended up being useful for. (The story is told that Schrödinger himself invented the Klein-Gordon equation even before his non-relativistic version, but discarded it because it didn’t do the job for describing the hydrogen atom. As my old professor Sidney Coleman put it, “Schrödinger was no dummy. He knew about relativity.”)

The Klein-Gordon and Dirac equations are actually not quantum at all — they are classical field equations, just like Maxwell’s equations are for electromagnetism and Einstein’s equation is for the metric tensor of gravity. They aren’t usually taught that way, in part because (unlike E&M and gravity) there aren’t any macroscopic classical fields in Nature that obey those equations. The KG equation governs relativistic scalar fields like the Higgs boson, while the Dirac equation governs spinor fields (spin-1/2 fermions) like the electron and neutrinos and quarks. In Nature, spinor fields are a little subtle, because they are anticommuting Grassmann variables rather than ordinary functions. But make no mistake; the Dirac equation fits perfectly comfortably into the standard Newtonian physical paradigm.

For fields like this, the role of “position” that for a single particle was played by the variable x is now played by an entire configuration of the field throughout space. For a scalar Klein-Gordon field, for example, that might be the values of the field φ(x) at every spatial location x. But otherwise the same story goes through as before. We construct a wave function by attaching a complex number to every possible value of the position variable; to emphasize that it’s a function of functions, we sometimes call it a “wave functional” and write it as a capital letter,

\Psi[\phi(x)].

The absolute-value-squared of this wave functional tells you the probability that you will observe the field to have the value φ(x) at each point x in space. The functional obeys — you guessed it — a version of the Schrödinger equation, with the Hamiltonian being that of a relativistic scalar field. There are likewise versions of the Schrödinger equation for the electromagnetic field, for Dirac fields, for the whole Core Theory, and what have you.

So the Schrödinger equation is not simply a relic of the early days of quantum mechanics, when we didn’t know how to deal with much more than non-relativistic particles orbiting atomic nuclei. It is the foundational equation of quantum dynamics, and applies to every quantum system there is. (There are equivalent ways of doing quantum mechanics, of course, like the Heisenberg picture and the path-integral formulation, but they’re all basically equivalent.) You tell me what the quantum state of your system is, and what is its Hamiltonian, and I will plug into the Schrödinger equation to see how that state will evolve with time. And as far as we know, quantum mechanics is how the universe works. Which makes the Schrödinger equation arguably the most important equation in all of physics.

While we’re at it, people complained that the cosmological constant Λ didn’t appear in Einstein’s equation (6). Of course it does — it’s part of the energy-momentum tensor on the right-hand side. Again, Einstein didn’t necessarily think of it that way, but these days we know better. The whole thing that is great about physics is that we keep learning things; we don’t need to remain stuck with the first ideas that were handed down by the great minds of the past.

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47 Responses to You Should Love (or at least respect) the Schrödinger Equation

  1. I have heard that Shrodinger’s equation is usted even by Bohmian Mechanics. Is that true?

  2. Hi Sean,

    Some underlying math talk, as consistent with “poetic naturalism” or “words can be defined scientifically”:

    What our favorite scientific descriptions have in common is additional multiplication,
    as in: “the outcome is given by the wave function squared”.

    Likewise (a suggested exercise, comparing square-root “rules”):

    We can label points as “imaginary numbers” that, when self-multiplied, produce square-root distances (in any potential direction); this defies the zero-origin convention of producing negative integers and distances like (-1).

    Synonymously (“quantizing gravity”, “accounting for inverse-square laws”): we can label any initial field as (+1): the square roots of positive one, divisible by their Cantorian-diagonal fields (roots of +2) and (roots of +3);

    Continuing to divide space; (diameter before radii): (+4) corresponds to (+1) at alternating points: predictions and observations. That’s it (“identifier mathematics” or “the rule of evidence”): let the ability to identify (square-root symbol) differentiate between complete opposites (using whatever symbols we choose). In other words, disproving solipsism isn’t beyond the scope of science (in terms of the minimal potential experiment):

    (prediction Alice at point Alice): everything (including point Alice) will not be annihilated.
    (prediction Bob at point Bob): everything (including point Bob) will be annihilated.
    (observation AliceBob): everything (including the distance AliceBob) was not annihilated.
    (theory AliceBob): everything excludes nothing in favor of any addition evidence (AliceBob).

    In other words (the limit of any potential observer): “one thing is not another”; all scientific equations can be reduced to square-root differentiation (and their squares); any other theory (as evidence) must deny everything (including itself, in motion, as self-contradictory).

  3. d says:

    >> Moshe

    *covariance* is not the same thing as *invariance*.

    (though it is true that schrodinger eq is more general than the non-rel, one-particle case usually discussed in textbooks)

  4. Peter Bilski says:

    Things becomes even more interesting ( one could say more complicated or less for me) if we assume the existing status of the system is resultant of the events going into opposite direction ( in time line) .
    Yours consideration always take into accounts our direction of the time events and neglect those going in opposite. Once you realize that our reality is actually resultant of the events simultaneously and continuously going into opposite direction, things becomes clear, or your understanding of Universe more complete. According to my supersymetry model our Universe is following this two opposite direction simultaneously. For the Big Bang to eternity…. and in opposite same time.

  5. Katrin Boeke-Purkis says:

    Imagine you are in a very dark cave — a Carlsbad Salt Dome — and your guide turns off all light emitting devices….now perceive these equations with your mind.

    “The task is…not so much to see what no one has yet seen; but to think what nobody has yet thought, about that which everybody sees.”
    Erwin Schrödinger

  6. Nathan Moynihan says:

    No one has mentioned the Parke-Taylor formula, a beautiful equation:

    [ latex ]A[1^-2^-3^+4^+] = \frac{\langle 12\rangle}{\langle 12\rangle\langle 23\rangle\langle 34\rangle\langle 41\rangle}[ /latex ]

  7. Daniel Kerr says:

    I feel like the confusion over the Schrödinger equation is sensible considering the equation’s history and that it really has two usages as a differential equation (QM) and a functional differential equation (QFT). Schrödinger’s own derivation builds up the Hamiltonian for a classical particle by considering the particle as a de Broglie wave. Through a modern lens, this treatment is seen as a differential equation of the coordinate representation of a wavefunction. Writing the equation in a more general, representation-independent form as a generator of time displacements via a unitary action on abstract Hilbert states, the result is equation 7. In quantum field theory when one identifies Hilbert Space vectors as functionals on field configurations, equation 7 remains mostly the same except the kets are understood to be different mathematical objects and the equation is now a functional differential equation dependent on the field configuration.

    Calling this last relationship the Schrödinger equation in the context of QFT has become commonplace but it’s not at all the framework in which the original Schrödinger equation was derived or even thought about. I don’t think it’s wrong to say that these field equations are not “the Schrödinger equation.” It’s also true to consider the Dirac equation as “a Schrödinger equation” with a different Hamiltonian. However one must distinguish the Dirac equation from the Dirac field equation; the latter is the quantization of the former when treated as a classical field equation. The Dirac equation itself has a purely quantum mechanical interpretation (though not without apparent paradoxes such as negative energies) as a “relativistic” Schrödinger equation which acts on spinors and not fermionic fields. Conversely one could treat the Schrödinger equation as a classical, scalar field equation and use second quantization to get a Schrödinger field which obeys the Schrödinger equation as written in equation 7. I feel this last case really highlights the confusion especially since both notions of the Schrödinger equation coincide for a Schrödinger field while they do not necessarily for other fields.

    Replacing states by operators and integrating over the Hamiltonian densities to get a Hilbert Space operator then calling the equations the same thing is really disorienting and I wouldn’t fault anybody for saying that the QFT version of the Schrödinger equation is really a completely different object than the Schrödinger equation in QM. Perhaps introducing the idea early on in QM curriculum that the Schrödinger equation is a general relationship between states and Hamiltonians is a good idea as it immediately dispels the notion that the Dirac equation might “replace” the Schrödinger equation and rather is just a specific instance of it. Going into anymore detail than that is too confusing in my opinion.

  8. vmarko says:

    Daniel Kerr,

    “Writing the equation in a more general, representation-independent form as a generator of time displacements via a unitary action on abstract Hilbert states, the result is equation 7.”

    This is possible in QFT because in QFT one assumes global Poincare symmetry (and flat spacetime), and the notion of a generator of global time displacements makes sense. Consequently in QFT one can write the Schrodinger equation.

    But when quantizing general relativity, the Poincare symmetry becomes a gauge symmetry, meaning that the action of the Hamiltonian (the time-translations generator) must vanish when acting on the Hilbert space vectors. This is a consequence of the localization of Poincare symmetry, in complete analogy wth QED where the localization of U(1) symmetry implies the Gupta-Bleuler condition.

    So in any ultimate theory of quantum gravity (or theory of everything, if you want to call it that way), the Schrodinger equation fails to be fundamental. At best, it tells you merely that the time-derivative of the wavefunctional is automatically zero, which is a fairly trivial statement. General covariance of the classical general relativity already tells us that time and space coordinates can be chosen arbitrarily, so the wavefunction cannot really depend on them in any meaningful way. The fundamental role of the Hamiltonian in QM thus gets completely wiped out in QG.

    Of course, people who are not experts in QG are usually not aware of this fact, and tend to ascribe more “weight” to the Schrodinger equation than it deserves, IMO.

    Best, 🙂
    Marko

  9. Moe says:

    @vmarko

    “Of course, people who are not experts in QG are usually not aware of this fact, and tend to ascribe more “weight” to the Schrodinger equation than it deserves, IMO.”

    I think the weight is given due to the fact that gravity is so weak in most of our normal experiences. If we could actually observe the relativity of time and space with our own eyes and unlearned brains, then we might not all have an intuition that doesn’t see the connections between spacetime and quantum mechanics. Or… we would be able to see the connections between entanglement and gravity more clearly.

  10. Daniel Kerr says:

    vmarko,

    I’m speaking only within established theories. Obviously without Poincare symmetry you don’t have a Lie Algebra that contains the Hamiltonian. However, it is speculative to claim that QG wouldn’t have poincare symmetry, it is possible to have “background Poincare symmetry” where gravitation is a field theory on that geometry. Quantization of this doesn’t work but there’s no reason to assume that some final theory wouldn’t have such a formulation. Alternatively, there might exist some symmetrical parameter which can be ascribed a dynamical role and a Schrödinger type equation would exist for this parameter. The semantic vagueness of the equation would be extended yet again like it was for QFT from QM.

  11. vmarko says:

    Daniel Kerr,

    “I’m speaking only within established theories..”

    So am I, and I consider classical general relativity to be fairly well established. 🙂 And it simply doesn’t have global Poincare symmetry. 😉

    But seriously, if you insist on keeping the global background Poincare symmetry, then yes, you do get to keep the Schrodinger equation as well. But this is going to be at odds with GR, where there is not even a single timelike Killing vector (in a generically curved spacetime) to define a global time-translation symmetry (let alone the full Poincare). Then your QG model would need to explain how and why do we get GR in the classical limit, how does the global Poincare invariance get lost in this limit, and why the standard model of cosmology (Lambda-CDM) explains all the observations so well while having no global time-translation symmetry. That’s the established/experimentally supported aspect of it.

    As far as theoretical/aesthetic aspects are concerned, by insisting on the background Poincare symmetry you are actually giving up the general principle of relativity, on which the whole Einstein’s GR is based. IOW, you are explicitly denying background independence of GR at the quantum level. This is also a question of motivation — which classical theory of gravity are you actually quantizing? It’s certainly not GR, but some alternative theory of gravity which is not background independent (and certainly not a very well established theory of gravity). This is an old and well-known open problem for example in string theory, where gravity is described as a spin-2 field in flat Minkowski spacetime (which has global Poincare symmetry), so diffeomorphism symmetry of classical GR is manifestly broken and nobody knows how (and if) it can be recovered at the quantum level, in the elusive “full” M-theory description.

    In general I do agree with what you said — it certainly is possible to have such a QG model. But, it would be very very ugly, ill-motivated from the GR point of view, it would have the practical problem to explain how do we recover GR in the classical limit, and why we do not see global Poincare symmetry in experiments, for example in cosmology and astrophysics. IMO, keeping the Schrodinger equation is not worth all that pain. 😉

    Btw, in his book/lectures on gravitation, even Richard Feynman had to admit (at the far end of the book) that treating gravity as a spin-2 field in flat spacetime isn’t enough, and one eventually needs to pass to full GR, curved spacetimes, etc. So IMO, global Poincare symmetry is a bad choice to stick to when quantizing gravity. Though I know some string theorists who would disagree, of course.

    Best, 🙂
    Marko

  12. Daniel Kerr says:

    I don’t think Sean’s point was made in the context of quantum gravity so my original answer does not hold in that context even if there is a possibility it could extend to that. None of this undermines his original point of the equation being foundational at least from a pedagogical point of view.

  13. vmarko says:

    Ok, I agree that your original comment was not aiming at QG, fair enough. Please don’t take my responses to you as criticisms of what you were saying, but rather as a suggestion that there’s more to say about the Schrodinger equation than just what QM and QFT say about it.

    Second, while I wouldn’t say I know what Sean’s point was, in the article he did write this:

    “So the Schrödinger equation is not simply a relic of the early days of quantum mechanics […] It is the foundational equation of quantum dynamics, and applies to every quantum system there is.”

    And my response was mainly to note that — no, it is *not* foundational, and it does *not* apply to every quantum system there is. Early in the comments Reader297 reminded us that it doesn’t apply to open quantum systems, which is basically everything except cosmology. And as I discussed at length above, one needs to jump through various hoops and bend over backwards to make it apply to general relativity and QG, which should describe that last remaining isolated system, cosmology.

    That’s the only issue I raised regarding Sean’s article. Of course, I agree that pedagogically the Schrodinger equation is an important concept to understand both in QM and QFT, but that Sean’s paragraph above is a complete overkill, is almost completely wrong, and is counterproductive even from a pedagogical point of view. So I fell compelled to react, for the sake of all the readers out there. 😉

    Best, 🙂
    Marko

  14. Marko,

    Totally. … Consider using “total(ly)” in place of “complete(ly)”. 😉

    –Ajit
    [E&OE]

  15. Daniel Kerr says:

    Open quantum systems are subsets of closed quantum systems, so any evolution equation in the open system could be expressed as a sort of “slice” from the evolution of the larger closed system. While practically one may never be able to completely characterize the close system, I feel like the open system case is still not a complete counterexample.

    As a separate line of criticism of Sean’s point I follow your logic and agree. I just wanted to be clear that I wasn’t advocating the generality to quantum gravity in my post. I only wanted to make the assumptions Sean did in order to provide some context about the confusion and justify considering them as separate equations within his paradigm.

  16. Robin says:

    Perhaps Archimedes Principle should be there (or his law of the lever) for its seminal importance, as it established the concept of describing nature by mathematical equations.

  17. KC Lee says:

    A possible stone for two birds: 1) QM textbook and 2) complex numbers.

    In their 2014 book “The Physics of Quantum Mechanics”, two professors at Oxford, Jame Binney and David Skinner, made a statement to the effect that nobody knows why complex numbers appeared in QM (See “Probability amplitudes” on p. 11).

    They think probability amplitudes set QM apart more so than does the commutation rule (which they could re-derive from purely classical considerations, p. 84).

  18. Paul Booker says:

    Nicely explained. Thanks for sharing.

  19. Lorenz says:

    Great article! The discussion of the quantum field wave function reminded me of Schwinger’s series “Theory of Quantized Fields”. I guess there are some differences to what is considered here, for example Schwinger is always considering the evolution of Fields from on spacelike hypersurface, but it seems to me related enough to point it out… additionally since i am not aware of many other sources about this topic. If somebody happens to know a good reference for this QFT wave function perspective, i would be glad to know!

    Besides that, i have a question regarding your assertion that the Dirac equation is perfectly fine in a classical setting as an equation of spin 1/2 particles. My problem is that a single classical spin 1/2 particle is unphysical in my understanding, since it does not transform in a reasonable way under rotations (only according to some projective representation).

  20. Kit Adams says:

    I am curious to see a Lorentz invariant field theory written in the form of equation 7. Does anybody know where to find an example?

    Presumably, in the context of “Space from Hilbert Space”, the t in equation 7 would be the “clock tick” driving a “simulation” of an emergent space time, not the variable rate, emergent local time determining the rate of processes within that space time, which, for example, has to slow down in the vicinity of regions of high energy density.

    The i in equation 7 is needed to make the evolution of the system unitary. Does unitarity imply that a “Space from Hilbert Space” type spacetime has to be a multi-verse comprised of all possible outcomes of all possible measurements?

    I have long been of the view that the “mysteries” of QM, i.e. wave/particle duality, EPR and the Bell inequality violations should be regarded as experimental evidence about the nature of spacetime, rather than being “interpreted away” as e.g. splitting universes or a result of the nature of consciousness, etc.
    A space time emerging from entanglement can perhaps explain these non-local effects in a natural way, but I suspect that a gravity related non-unitary process something like Roger Penrose’s gravitational objective reduction (OR) will also be needed (and will arise naturally) in a successful theory of spacetime. Is there any way such a thing could arise from equation 7?

    At least Penrose’s OR is potentially experimentally testable in the not-too-distant future.

  21. John says:

    Physicist Richard Feynman referred to Euler’s Identity as “our jewel” , “the most remarkable formula in mathematics”. Maybe he etched it somewhere on his desk, which is now your desk.