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

47 Comments / Science

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 Comments

47 thoughts on “You Should Love (or at least respect) the Schrödinger Equation”

  1. Richard Casey

    Not only would I generally agree with this set of equations as my favorites, but I have them as screensavers to remind me of their fundamental importance in scientific inquiry. Thanks for sharing.

  3. BobC

    As an engineer I’ve always enjoyed the elegance and power of Maxwell’s Equations, but when it comes to practical matters (making stuff that works), few equations eclipse Ohm’s Law and Kirchoff’s Laws (KVL and KCL), and few tools eclipse the Laplace and Fourier transforms.

  4. John Anderson

    I love lists like this. Similar fun with Robert Crease’s book “The Great Equations”, and Wm. Dunham’s book “Journey through Genius” or the video course “Great Thinkers Great Theorems”.

    Forty years after I studied Q.M. in college I got a proof from CU/Boulder (really just a justification of Schrödinger’s equation); I said “Wow, it’s not just a postulate; it makes some sense!”

    Weird Al Yankovic made music history by having Schrödinger’s equation as part of his video backdrop in “White and Nerdy”.

  5. Fred Gray

    Wow! Made my day. I just realized that I’m really stupid, in the grand scheme of things. Bummer.

  6. Reader297

    Sean–

    Great post! I agree about respecting the Schrodinger equation, and not confusing it with its special case for a nonrelativistic system of particles, nor with the common idea that somehow the Dirac equation or the Klein-Gordon equation generalize the Schrodinger equation.

    However, your post has an important couple of typos or oversights. If one is going to go to all the trouble of trying to rectify popular misconceptions, one should be sure not to instill new ones!

    You write “Every quantum system obeys a version of the Schrödinger equation; it’s completely general.” You later also write “It is the foundational equation of quantum dynamics, and applies to every quantum system there is… 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.”

    These statements are not correct. They apply only in the special case of a *closed* quantum system. For a *closed* system that begins and remains in a pure state, the Schrodinger equation expresses conservation of information with time, with the consequence in particular that the system’s Von Neumann entropy is and remains exactly zero.

    An *open* quantum system, however, does not generically begin or end in a pure state, does not generically conserve information with time, and does not generically satisfy the Schrodinger equation. After all, without being in a pure state, there isn’t a state vector or wave function available even in principle to satisfy the Schrodinger equation in the first place!

    The dynamics of an open system is, at best, expressible in terms of the system’s density matrix. With certain assumptions and approximation, open quantum systems evolve according to a linear CPTP mapping, an example of which is the Lindblad quation (https://en.wikipedia.org/wiki/Lindblad_equation), which assumes linearity and homogeneity in time, both of which are always just approximations in practice.

    Open systems certainly aren’t just a curiosity. All systems apart from the entire universe are generically at least a little bit open. So not only does the Schrodinger equation not describe general systems, it doesn’t really describe *any* systems at all, apart from the universe as a whole. Even more importantly, any system undergoing a measurement process is at least temporarily very much open, and, of course, an ideal measurement process without postselection can be written as a linear CPTP mapping, namely, the Von Neumann-Luders projection.

    For a totally general quantum system and without making approximations or assumptions, it isn’t even the case that the system’s evolution is necessarily linear.

  7. Moshe

    Surprising how many people who should know better still talk about “relativistic quantum mechanics”, even when teaching graduate classes, and consequently just how much unconfusing you have to do when teaching QFT. One more common confusion is thinking the Schrodinger equation is non-relativistic because it involves time derivatives but not spatial ones (of course both sides of the equation transform the same way under Lorentz transformations, so the equation is perfectly Lorentz invariant, even if it is not phrased as a statement about Lorntz scalers)

  9. Richard Gaylord

    “we don’t need to remain stuck with the first ideas that were handed down by the great minds of the past.”. it’s comforting to know that string theory will ‘go away’ eventually. unfortunately, in the words of Max Planck ““Science advances one funeral at a time.”

  10. vmarko

    Sean,

    I have to say that I agree with Reader297 on this. The Schrodinger equation works for isolated systems only, so basically it’s just an approximation rather than a fundamental equation of nature.

    Reader297,

    “So not only does the Schrodinger equation not describe general systems, it doesn’t really describe *any* systems at all, apart from the universe as a whole.”

    Not only that, but the Schrodinger equation doesn’t apply even to the universe as a whole. In general relativity (and basically all physical systems which have diffeomorphism symmetry) the Hamiltonian of the system is a linear combination of the constraints, and is therefore equal to zero. So the Schrodinger equation for the whole universe just tells you that the total wavefunctional doesn’t depend on time. IOW, there is no equation to solve there, it’s trivial.

    So I’d say that the Schrodinger equation is far from being fundamental in any respect. It’s not relevant for any realistic physical system, unless you introduce approximations.

    🙂
    Marko

  11. Bob Zannelli

    I think it’s ironic that Schrodinger was not very pleased with what this equation is really telling us about nature. Its “true” meaning was made clear by Max Born, perhaps one of the most overlooked of the great physicists who gave us the quantum revolution. Like Einstein, Schrodinger was never reconciled with what he has discovered.

  12. masmadera

    Eq.(4) is only important to the extent that one adds the second principle (the entropy always increases or remains the same, defining the equilibrium). Otherwise is like writing how you compute the energy without saying why the energy is important (time invariance). But Thermo seems to be getting old fashion and somehow few people think now that Delta S > 0 is a beautiful equation.

    I fully disagree with commenters downgrading the Schrödinger equation. I agree that it is incomplete, in the general framework of Quantum Mechanics, without invoking Born’s rule. But then, you can only pick 7 equations!

  13. Moe

    In your last post you said that quantum mechanics says that the real world is best described by a wave function, an element of Hilbert space, evolving through time.

    Do you believe that the total wave function has a time dependency?

    I find it very hard to understand how space and time can be so intermingled in Relativity, yet seem to seperate so much in quantum mechanics. Even in considering gravity emerging from entanglement entropy, time seems to have some special position seperate from the other dimensions.

  14. Carl

    Sean,

    What a great post and concise explanation of the complex relationships of the relevant equations. This should be included in every “Introduction to QM” class.

    “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.”

    I totally agree about your assessment of the way “traditional” QM is usually introduced. That was my introduction to QM.

    Thanks again for a great post.

    Carl G.
    =====

  15. Sean Carroll

    Moe– I honestly don’t know whether the true wave function of the universe is time-dependent or not. It’s very possible that the apparent time evolution of the universe is an emergent phenomenon, just like space. We’ll have to see.

    It’s true that the Lorentz symmetry of relativity seems to put time and space on an equal footing, which they don’t seem to be from the perspective of the Schrödinger equation. But again, Lorentz invariance could just be an emergent approximation.

  16. Joseph Salmon

    Please let me encourage another “TEDD Radio Hour” talk on this and other ambitious projects.

  17. Moe

    @Sean

    Thanks for those thoughts.

    Maybe I haven’t caught this before.

    Are you in favor of the idea that the universe is ‘just information’?

    As in… space and gravity just energe from entanglement entropy, and time also emerges from entropy?

    Then the whole universe could just be described by bits of information?

    Potentially?

  19. Left Coast Bernard

    Prof. Carroll,
    Please recommend a (graduate level) quantum physics book that explains the ideas about quantum theory correctly as you describe in this interesting post.
    Also, I wonder if you might comment on the apparent essential place of complex functions and numbers in quantum theory. After all, any experimental result or observation will be a rational number. Physics theories usually deal in real numbers and real functions. Schrodinger’s earliest papers state results as the real part or the imaginary part of the wave function, but now physicists see the wave function’s complex nature as central, and not just a matter of computational convenience as complex functions are, say, in electrical engineering. Computers, on the other hand, have a finite number of possible numbers, with a maximum, a minimum, and a limited precision.

  20. Sean Carroll

    I don’t really have a favorite grad-level QM book, I’m afraid. About complex numbers, people have speculated on the deeper significance there, but it’s not something I’ve personally thought about.

  21. Gabriel

    Great post, I’ve wondered for awhile what the general Schroedinger equation encompasses. I’ve been reading The Big Picture and enjoying it very much. Just wondering if a Schroedinger equation for the core theory has been written down by anyone or if you were going to possibly post one? Also, curious why the yang-mills equations didn’t make the list.

  22. Julio Siqueira

    .
    As an incurable layman in physics ( 🙁 ), I try erratically and clumsily to grasp some inklings of physic’s insights from writings of people like Sean Carroll, both in books and in blog posts (articles) like this one that he just wrote. And often I find perplexing oppositions like the ones expressed by the phrases below:
    .
    [Sean Carroll] And as far as we know, quantum mechanics is how the universe works.
    .
    [Reader297] So not only does the Schrodinger equation not describe general systems, it doesn’t really describe *any* systems at all, apart from the universe as a whole.
    .
    [vmarko] Not only that, but the Schrodinger equation doesn’t apply even to the universe as a whole. So I’d say that the Schrodinger equation is far from being fundamental in any respect.
    .
    Now, you enlighted ones, imagine how we humble laymen feel when facing these perplexing different scientific views…
    .
    Best Wishes,
    Julio Siqueira
    juliocbsiqueira2012@gmail.com
    http://www.criticandokardec.com.br/criticizingskepticism.htm
    —————————————————————————————————

  23. John B

    A part of me dies a little bit inside every time you fight for the MWI, because I have always heard that the Schrodinger equation implies that quantum uncertainty is fundamental.

  24. Moe

    @John B

    Quantum uncertainty is not in the Schrödinger equation. It is completely deterministic. Uncertainty only exists when measurements are taken. MWI allows these to both be true. The uncertainty only exists within particular branches that seem to be unique, from the standpoint of the observers within them. Taken as a whole, MWI results directly from the Schrödinger equation, without ever needing to collapse the wavefunction.

  25. MassiveFermion

    Hi Sean
    Thanks for the nice post
    I have a question about it though.
    When you try to pursue a parallel path with non-Rel. QM, surely it seems very plausible to think that the square of the absolute value of the wave-functional gives the probability for the field to be in the configuration that was given to the wave-functional as the input. But when I think about it, it seems to me that its useless to the extent that I wonder why such an identification should be made.
    There are two reasons for what I said:
    1) Its not possible, even in theory(I think), to conduct an experiment that gives us enough information to find out which configuration the field actually is in.
    2) Its not possible, even in theory, to have an ensemble of identically prepared fields.
    So I’m wondering whether it even makes sense to have a functional formulation of QFT!
    What do you think?

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