I grip amongst these ii points.
You could wonder why Sean Carroll hasn't written downwardly the uncomplicated judgement inwards the offset paragraph of this weblog ship as well as instead, added roughly 3 pages of redundant text. Well, it's because he needed to move yesteryear along the per centum of misconceptions as well as distortions good higher upwards 50%, just similar inwards almost all of his texts almost quantum mechanics.
I volition live to a greater extent than specific almost the complaints but permit me laid about amongst his listing of his "seven favorite equations":\[
\eq{
{\bf F} &= m{\bf a}\\
\partial L/\partial {\bf x} &= \partial_t ({\partial L}/{\partial {\dot {\bf x}}})\\
{\mathrm d}*F &= J\\
south &= k \log W\\
ds^2 &= -{\mathrm d}t^2 + {\mathrm d}{\bf x}^2 \\
G_{ab} &= 8\pi G T_{ab}\\
\hat{H}|\psi\rangle &= i\partial_t |\psi\rangle
}
\] OK, overnice enough, specially if nosotros tin upgrade Carroll's inferior techniques to write \({\rm\LaTeX}\) equations to the full-fledged MathJax. I wouldn't write the exact same listing – e.g. because there's no jurist theory, string theory, or supersymmetry inwards the listing – but let's non utter over advanced topics.
There are at to the lowest degree the next occupation amongst the presentation as well as "framing" of the Schrödinger equation equally offered yesteryear Sean Carroll – as well as lots of other sources of superficial popular scientific discipline out there:
- Looks classical: the equation of quantum mechanics is written equally if it were roughly other classical equation
- The time-dependent moving ridge business office is made to human face unavoidable inwards QM: this is just an artifact of the formalism
- Schrödinger's misunderstanding of the pregnant of the moving ridge business office is beingness obscured: solely Max Born did this of import task as well as got his well-deserved Nobel prize for that
- Schrödinger is incorrectly credited: amongst the generalizations of the equation for other Hamiltonians \(H\)
- Sometimes there's no \(H\): it's beingness incorrectly suggested that the Hamiltonian \(H\) must live well-defined inwards whatever predictive quantum mechanical theory
The qualitative saltation is beingness obscured
Carroll wrote his 7 favorite equations, starting from \(\vec F = m\vec a\). You may regard that all of them are roughly mathematical equations that are relevant inwards physics. The offset half dozen of these equations are equations relevant inwards classical i.e. non-quantum physics as well as they're increasingly mathematically abstract, advanced, or sophisticated. One is gradually adding fields as well as non just positions of particles, perchance statistical entities such equally the entropy, or the metric tensor; ordinary differential equations are beingness superseded yesteryear partial ones, as well as thus on.
One could live easily tempted to extrapolate this tendency as well as say that the progress inwards physics is only a path towards increasingly to a greater extent than abstract classical equations that define the dynamical laws of physics. Unfortunately, that's indeed how the people brainwashed yesteryear the popular scientific discipline – mostly included Carroll himself – sympathise the role of the seventh, Schrödinger equation.
However, that's completely wrong. The measurement from equations such equally Einstein's equations (sixth inwards the listing of seven) to the Schrödinger equation is a measurement of a completely different character than whatever of the previous steps. Quantum mechanics is not roughly other classical theory amongst somewhat to a greater extent than complex equations for a somewhat to a greater extent than complex laid of degrees of freedom.
Instead, quantum mechanics is a qualitatively novel theory – or a novel framework – that doesn't alter the dynamical equations equally much equally it changes the interpretation of the mathematical symbols i.e. the human relationship betwixt the mathematical objects as well as the empirical observations. Quantum mechanics genuinely differs from classical physics yesteryear its observables' beingness non-commutative operators. For that reason, eigenvalues accept to live interpreted equally possible values as well as predictions accept to live made inwards a probabilistic way yesteryear squaring complex probability amplitudes.
H5N1 to a greater extent than conceptually right way to explicate quantum mechanics involves the original, Heisenberg motion painting of quantum mechanics. In the Heisenberg picture, the dynamical equations such equally the offset i (or others that involve fourth dimension derivatives of dynamical degrees of freedom) tin live pretty much kept fixed. You just add together hats, upgrade the coordinates as well as momenta to operators, as well as postulate that operator equations accept to supersede the master equations for \(c\)-numbers.
In particular, the equation \(\vec F = m\vec a\) is just fine inwards quantum mechanics if you lot only add together the hats.
Focus on the picture, non the essence
I am basically going to brand a related dot which I decided to frame equally an independent one. I just said that people are oft made to believe that the moving ridge business office is roughly other classical land – analogous to the electromagnetic or metric land – as well as the Schrödinger equation is roughly other classical dynamical equation – analogous to Maxwell's or Einstein's equations.
In the novel weblog post, Carroll does nada to counter this misconception. In many other texts, he straight helps inwards the propagation of this deep misunderstanding.
One argue why many laymen purchase this untruth is the fact that the Schrödinger equation is notwithstanding a differential equation. Differential equations are "hard enough" as well as i time people are forced to remember almost the differential equation, they don't accept whatever spare release energy to remember almost anything else. But the other things, the pregnant of the symbols as well as their relationships to the observations, are much to a greater extent than of import than the mathematical cast of the equation.
In fact, it is non necessary inwards quantum mechanics to bargain amongst the time-dependent moving ridge business office at all. Quantum mechanics may live formulated inwards the Heisenberg motion painting – the modern reformulation of the "matrix mechanics", the offset way to write downwardly the rules of quantum mechanics equally understood inwards the pioneering papers yesteryear Werner Heisenberg. In this picture, \(\ket\psi\) is independent of time. Instead, it's the operators \(\hat x(t),\hat p(t)\), as well as others that are time-dependent. And the equations for their fourth dimension derivatives pretty much re-create the corresponding equations inwards classical physics. You just add together hats. Expectation values may notwithstanding live computed inwards the commons way. Probabilities may live computed equally the expectation values of projection operators (which are time-dependent here, much similar almost all operators).
And yes, i tin formulate quantum mechanics without whatever differential equations containing time-derivatives. Richard Feynman flora the consummate way to calculate all predictions inwards quantum mechanics "directly", without solving whatever differential equations (for functions of time). You just amount complex amplitudes over all histories or trajectories. The complex outcome is the probability aAmplitude for i desired development from a known initial solid soil to a possible finite state. You foursquare the absolute value as well as obtain the probability. Nothing else than the probabilities (or things that are manifestly functions of them) tin live calculated according to the full general principles of quantum mechanics.
The excessive focus on the Schrödinger motion painting – as well as the fact that popular scientific discipline as well as many textbooks suppress the other pictures as well as formulations – is roughly other driver that encourages the students as well as laymen to strengthen their belief that the Schrödinger equation is just roughly other classical equation inwards a sequence of increasingly mathematically complex equations of classical physics.
Schrödinger himself didn't know the right physical pregnant of the symbols
Schrödinger himself idea that the moving ridge business office is a "dissolved electron" which is spread over the infinite much similar butter is spread over bread. In roughly texts, he made this dot rather explicitly. Werner Heisenberg as well as others already understood that the moving ridge business office was a probability aAmplitude moving ridge as well as they were explicitly critical of Schrödinger. But he has never understood the right physical pregnant of the moving ridge business office thus all his papers ever remained partly wrong as well as partly ill-defined.
Just to live sure, the papers weren't bad. He was real proficient inwards dealing amongst equations of mathematical physics as well as he wasn't trying to spread his misconceptions everywhere. If you lot read the newspaper Schrödinger 1926c which is helpfully inwards English linguistic communication as well as non inwards High German (as a modern physicist, Schrödinger sent it to The Physical Review), you lot volition accept a rather difficult fourth dimension to discovery the explicitly wrong claims almost the "visualizability" of the moving ridge business office etc. But these opinions are implicitly in that location as well as he has repeatedly made the wrong statements explicitly.
Generalizations to other Hamiltonians \(H\) aren't genuinely his invention
This is roughly other mostly historical or sociological particular but you lot may see it equally an of import one, too. Carroll claims that the Schrödinger equation may live used fifty-fifty amongst the Hamiltonian \(H\) that defines the total release energy e.g. of all fields inwards the Standard Model. If you lot do so, the Schrödinger equation correctly describes the development of all observable things at the accuracy of the Standard Model – which, except for the absence of gravity, describes everything nosotros accept ever safely observed inwards key physics.
Well, if you lot human face at the Schrödinger 1926c paper, you lot volition regard that what Schrödinger has actually discovered was really just the "edition" of the equation named later him that deals amongst non-relativistic mechanics. Carroll complains that people, when asked to write downwardly the Schrödinger equation, ordinarily write\[
\left[-\frac{1}{\mu^2}\frac{\partial^2}{\partial x^2} + V(x)\right]|\psi\rangle = i\partial_t |\psi\rangle
\] Indeed, nosotros oft similar to exercise the term "the Schrödinger equation" to a greater extent than generally. I exercise it to a greater extent than generally, too. On the other hand, when you lot read Schrödinger's actual papers, you lot volition regard that he's never written downwardly the full general equation. He genuinely derived the special equation higher upwards – just equally indicated yesteryear the people whom Carroll criticizes. He deduced this equation yesteryear roughly to a greater extent than convoluted thoughts almost the de Broglie moving ridge inwards the presence of external potentials.
Schrödinger wasn't thinking quantum mechanically as well as to a greater extent than oft than non almost the new, quantum mechanical framework for physics. And that was the principal argue why he wasn't able to move yesteryear beyond the instance of non-relativistic mechanics fifty-fifty though he arguably had the mathematical skills to do so. What happened?
Erwin Schrödinger genuinely did fully realize that the equation higher upwards is just a non-relativistic equation. He knew that it meant a limitation, he knew that relativity was right, he knew what relativity demanded, thus he knew that a relativistic version of the equation was needed. But what he ended upwards amongst was genuinely the Klein-Gordon equation\[
\left[ \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} + m^2 \right] \Psi(\vec r,t) = 0.
\] He genuinely tried to exercise this equation to "solve the hydrogen atom again". He got roughly results that disagreed amongst the experiments thus equally a well-behaved physicist, he threw the equation to the trash bin. Later, the equation was published yesteryear Klein, Gordon, as well as Fock. Even though Schrödinger has beaten them equally a faster mathematician, the equation never carries his get upwards because Klein, Gordon, as well as Fock scooped him inwards the journals.
Now, if Schrödinger had known almost the "general Schrödinger equation amongst a full general Hamiltonian \(H\)", he wouldn't fifty-fifty examine to apply the equation to a one-particle occupation of the hydrogen atom. Why? Because he would know that the Klein-Gordon equation higher upwards only isn't an illustration of the Schrödinger equation. So it cannot live considered a generalization of his non-relativistic mechanics Schrödinger equation.
Why the Klein-Gordon equation isn't a generalization of the non-relativistic Schrödinger equation? Because all full general Schrödinger equations accept to live equations for a complex moving ridge function; as well as they accept to live first-order differential equations inwards time. However, the minimal Klein-Gordon equation is an equation for a real, non complex field; and, to a greater extent than importantly, every Klein-Gordon equation is a second-order differential equation inwards time.
So it's only not whatever variety of a Schrödinger equation or its generalization! You can't translate it inwards this way.
As you lot know, the right relativistic Schrödinger-like equation for the electron was flora yesteryear Paul Dirac. It was the Dirac equation as well as straight off described the electron's spin (and antiparticles), too. The existent argue why Dirac was able to do this affair wasn't that Schrödinger was incapable of learning the spinors. The principal argue was that dissimilar Schrödinger, Dirac understood the pregnant of the symbols inwards the non-relativistic Schrödinger equation as well as the novel framework of quantum mechanics inwards general.
In particular, Dirac knew that fifty-fifty the "general" Schrödinger equation – fifty-fifty i for a relativistic hydrogen atom – has to live a first-order equation inwards time. That way as well as Dirac (but non Schrödinger) knew that it meant that the Klein-Gordon equation was at most a candidate for a novel equation governing a classical field, non a direct replacement for the Schrödinger equation. He was searching for a Hamiltonian operator that would automatically obey \(H^2 = (\vec p)^2 + m_0^2 \) as well as was forced to consider the spinors as well as the Dirac gamma matrices. Schrödinger wasn't solving this occupation at all because he wasn't fully aware of the fact that what he needed was definitely a first-order equation. In other words, Schrödinger wasn't solving the same fantabulous practise equally Dirac because Schrödinger didn't sympathise the full general equation that nosotros currently telephone band the Schrödinger equation (for roughly full general \(H\)). We credit him amongst something he clearly misunderstood!
It's natural to exercise the master get upwards for the of import generalization but it's historically misleading.
Quantum mechanical theories may be fifty-fifty if in that location is no \(H\)
The last, to a greater extent than advanced dot I wishing to brand is that it is non fifty-fifty truthful that the existence of a well-defined Hamiltonian operator \(H\) is needed for a quantum mechanical theory to exist. In the text above, I wrote that nosotros tin brand quantum mechanical prediction without ever using or solving the full general Schrödinger equation. We tin switch to the Heisenberg motion painting or to Feynman's approach inwards damage of the path integrals.
However, inwards those sections, you lot could notwithstanding assume that it's ever inwards regulation possible to seat the Hamiltonian – as well as to switch dorsum to the Schrödinger equation if you lot wishing to.
That's non genuinely the case. There may be quantum mechanical theories that are fully predictive but they don't give you lot whatever prescription for \(H\) at all. Nevertheless, they may predict what volition happen.
Quantum theories including gravity as well as the spacetime diffeomorphisms largely autumn into this category. String theory is the most well-defined illustration – and, for non-trivial reasons, the solely fully well-defined example, but that's roughly other discussion. If you lot wishing to move yesteryear along the diffeomorphism symmetry of full general relativity manifest at to the lowest degree to roughly extent, you're prevented from defining the Hamiltonian.
Note that the Hamiltonian is a generator of fourth dimension translations, \(t\to t+\delta t\), but such translations solely brand feel if you lot know what the corresponding fourth dimension coordinate \(t\) is. But a defining conceptual fact almost full general relativity is that in that location exists no preferred choice of the coordinates \(t,x,y,z\). You may pick almost whatever business office \(t'(t,x,y,z)\) of your master spacetime coordinates as well as telephone band \(t'\) your novel fourth dimension coordinate! There volition be corresponding variations of \(t'\) as well as the novel Hamiltonian operator \(H'\) that generates them.
In fact, if you lot appreciate the condition of these redefinitions, you lot volition realize that these diffeomorphisms or coordinate transformations are conceptually jurist symmetries – much similar Yang-Mills symmetries inwards Abelian or non-Abelian jurist theories – as well as the physical states accept to invariant nether these symmetries. It also way that physical states accept to live annihilated yesteryear the generators of such symmetries such equally \(H\). (Symmetry transformations that alter the fields fifty-fifty inwards the asymptotic regions at infinity may live given an exemption; these transformations' generators such equally the ADM release energy may live nonzero i.e. don't accept to annihilate the physical states.)
In particular, the equation \(H\ket\psi = 0\) is "true" inwards roughly full general formulation of quantum gravity as well as it is known equally the Wheeler-DeWitt (WdW) equation. Only approximate ways to formulate quantum gravity inwards damage of this equation are known.
However, fifty-fifty inwards the absence of a fully well-defined mathematical realization of the WdW equation, it's notwithstanding truthful that consistent quantum gravity theories to a greater extent than oft than non don't wishing to give you lot a clear prescription for the Hamiltonian \(H\). For example, covariant perturbative string theory straight produces the S-matrix of development probability amplitudes – those from \(t=-\infty\) to \(t=+\infty\). They're equivalent to on-shell Green's functions inwards QFT only. There are no overnice calculable formulae for off-shell Green's functions inwards string theory or other gravitating quantum mechanical theories. That's related to the non-existence of completely gauge-invariant local quantities inwards diffeomorphism-invariant theories. Even the scalar curved shape \(R(x,y,z,t)\) fails to live gauge-invariant because the jurist symmetries alter the values of \(x,y,z,t\) which way that they map this scalar to the scalar at roughly other point, \(R(x',y',z',t')\).
This disputation isn't "quite" truthful because if you lot gauge-fix the diffeomorphism symmetry, most famously yesteryear going to the light-cone jurist inwards string theory, it becomes possible to pull the development inwards damage of "one 2nd to another" steps. And the appropriate light-cone Hamiltonian, the factor of the vector \(P^-\) that evolves your solid soil of strings from i null while to another, becomes well-defined. Green as well as Schwarz loved to exercise this "light-cone jurist string land theory" inwards the early on 1980s. It was solely well-defined perturbatively but my as well as DVV's matrix string theory provides us amongst a total non-perturbative Definition of this development of strings from i 2nd to another. (The BFSS model that does the same task for the stringless 11-dimensional M-theory boundary was flora earlier matrix string theory.)
But those formulations of string theory are possible solely amongst roughly extra piece of work – gauge-fixing as well as efforts to discovery novel equations. If people had non flora the light-cone jurist formulations of string theory, it would notwithstanding live truthful that they tin brand infinitely many arbitrarily accurate predictions inwards string theory – the S-matrix – as well as the Hamiltonian as well as all equations subject on it would live completely circumvented.
The absence of a Hamiltonian may live encountered inwards other situations, too. Quite typically, nosotros write the Hamiltonian equally a business office or functional of roughly other observables. However, these observables are oft operators obtained yesteryear "adding hats" on roughly classical quantities. That's possible when roughly classical quantities be – i.e. when the quantum mechanical theory has a classical limit. This is non ever the case, either. Sometimes, fifty-fifty inwards non-gravitational theories, the Hamiltonian may formally be but in that location is no helpful "explicit formula" for such a Hamiltonian. Examples include roughly two-dimensional (but also six-dimensional) conformal land theories.
OK, to summarize:
There are diverse either explicitly wrong misconceptions or "encouragements to remember conceptually incorrectly" that are beingness spread inwards the get upwards of the Schrödinger equation. The misleading analogies betwixt the equation amongst the equations of classical physics, the focus on the particular cast of the Schrödinger equation, as well as the emphasis on the Schrödinger picture, as well as the excessive advertisement of the objects that the equation depends upon may live classified equally of import but non the solely reasons why thus many people goal upwards misunderstanding quantum mechanics – what it means, what it says, as well as inwards what feel it genuinely differs from the former classical physics.
So fifty-fifty though I grip amongst Carroll's ii well-defined assertions – that the equation is of import inwards physics; as well as it is applicable exterior non-relativistic mechanics of particles – I discovery his efforts to brand the readers (and himself) to a greater extent than deeply immersed inwards the specific features of this equation counterproductive. What people demand to larn quantum mechanics correctly is to suppress the dependence of their noesis on this equation as well as on the Schrödinger picture.
The Schrödinger equation isn't what gives the quantum flavour to quantum mechanics. The nonzero commutators or, equivalently, the doubtfulness regulation is the actual spice that makes the whole difference. Promoters of superficial popular scientific discipline including Sean Carroll can't "taste" this infinite because they mostly misunderstand quantum mechanics themselves.