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2022-11-05

Schrodinger's equation (explanation to non physicist)
For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the explanation. However I'm not so stupid to think I can understand this landmark without some years of physics. I'll just be happy to be able to read the equation and recognize it in its various forms.
To be more precise, here are my questions.
Hyperphysics tell me that Shrodinger's equation "is a wave equation in terms of the wavefunction".
1. Where is the wave equation in the most general form of the equation?
$\mathrm{i}\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }=H\mathrm{\Psi }$
I thought wave equation should be of the type
$\frac{{\mathrm{\partial }}^{2}}{{\mathrm{\partial }}^{2}t}u={c}^{2}{\mathrm{\nabla }}^{2}u$
It's the difference in order of of derivation that is bugging me.
2. Can somebody show me the passages in a simple (or better general) case?
3.I think this questions is the most difficult to answer to a newbie. What is the Hamiltonian of a state? How much, generally speaking, does the Hamiltonian have to do do with the energy of a state?
4.What assumptions did Schrödinger make about the wave function of a state, to be able to write the equation? Or what are the important things I should note in a wave function that are basilar to proof the equation? With both questions I mean, what are the passages between de Broglie (yes there are these waves) and Schrödinger (the wave function is characterized by)?
5.It's often said "The equation helps finds the form of the wave function" as often as "The equation helps us predict the evolution of a wave function" Which of the two? When one, when the other?

artirw9f

You should not think of the Schrödinger equation as a true wave equation. In electricity and magnetism, the wave equation is typically written as
$\frac{1}{{c}^{2}}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^{2}}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}}$
with two temporal and two spatial derivatives. In particular, it puts time and space on 'equal footing', in other words, the equation is invariant under the Lorentz transformations of special relativity. The one-dimensional time-dependent Schrödinger equation for a free particle is
$\mathrm{i}\hslash \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=-\frac{{\hslash }^{2}}{2m}\frac{{\mathrm{\partial }}^{2}\psi }{\mathrm{\partial }{x}^{2}}$
which has one temporal derivative but two spatial derivatives, and so it is not Lorentz invariant (but it is Galilean invariant). For a conservative potential, we usually add $V\left(x\right)\psi$ to the right hand side.
Now, you can solve the Schrödinger equation is various situations, with potentials and boundary conditions, just like any other differential equation. You in general will solve for a complex (analytic) solution $\psi \left(\stackrel{\to }{r}\right)$: quantum mechanics demands complex functions, whereas in the (classical, E&M) wave equation complex solutions are simply shorthand for real ones. Moreover, due to the probabilistic interpretation of $\psi \left(\stackrel{\to }{r}\right)$, we make the demand that all solutions must be normalized such that $\int |\psi \left(\stackrel{\to }{r}\right){|}^{2}dr=1$. We're allowed to do that because it's linear (think 'linear' as in linear algebra), it just restricts the number of solutions you can have. This requirements, plus linearity, gives you the following properties:
1. You can put any $\psi \left(\stackrel{\to }{r}\right)$ into Schrödinger's equation (as long as it is normalized and 'nice'), and the time-dependence in the equation will predict how that state evolves.
2.If $\psi$ is a solution to a linear equation, $a\psi$ is also a solution for some (complex) $a$. However, we say all such states are 'the same', and anyway we only accept normalized solutions ($\int |a\psi \left(\stackrel{\to }{r}\right){|}^{2}dr=1$). We say that solutions like $-\psi$, and more generally ${e}^{i\theta }\psi$, represent the same physical state.
3. Some special solutions ${\psi }_{E}$ are eigenstates of the right-hand-side of the time-dependent Schrödinger equation, and therefore they can be written as
$-\frac{{\hslash }^{2}}{2m}\frac{{\mathrm{\partial }}^{2}{\psi }_{E}}{\mathrm{\partial }{x}^{2}}=E{\psi }_{E}$
and it can be shown that these solutions have the particular time dependence ${\psi }_{E}\left(\stackrel{\to }{r},t\right)={\psi }_{E}\left(\stackrel{\to }{r}\right){e}^{-iEt/\hslash }$. As you may know from linear algebra, the eigenstates decomposition is very useful. Physically, these solutions are 'energy eigenstates' and represent states of constant energy.
4. If $\psi$ and $\varphi$ are solutions, so is $a\psi +b\varphi$, as long as $|a{|}^{2}+|b{|}^{2}=1$ to keep the solution normalized. This is what we call a 'superposition'. A very important component here is that there are many ways to 'add' two solutions with equal weights: $\frac{1}{\sqrt{2}}\left(\psi +{e}^{i\theta }\varphi \right)$ are solutions for all angles $\theta$, hence we can combine states with plus or minus signs. This turns out to be critical in many quantum phenomena, especially interference phenomena such as Rabi and Ramsey oscillations that you'll surely learn about in a quantum computing class.
Now, the connection to physics.
1. If $\psi \left(\stackrel{\to }{r},t\right)$ is a solution to the Schrödinger's equation at position $\stackrel{\to }{r}$ and time $t$, then the probability of finding the particle in a specific region can be found by integrating $|{\psi }^{2}|$ around that region. For that reason, we identify $|\psi {|}^{2}$ as the probability solution for the particle.
We expect the probability of finding a particle somewhere at any particular time t. The Schrödinger equation has the (essential) property that if $\int |\psi \left(\stackrel{\to }{r},t\right){|}^{2}dr=1$ at a given time, then the property holds at all times. In other words, the Schrödinger equation conserves probability. This implies that there exists a continuity equation.
2. If you want to know the mean value of an observable A at a given time just integrate

where $\stackrel{^}{A}$ is the linear operator associated to the observable. In the position representation, the position operator is $\stackrel{^}{A}=x$, and the momentum operator, $\stackrel{^}{p}=-i\hslash \mathrm{\partial }/\mathrm{\partial }x$, which is a differential operator.
The connection to de Broglie is best thought of as historical. It's related to how Schrödinger figured out the equation, but don't look for a rigorous connection. As for the Hamiltonian, that's a very useful concept from classical mechanics. In this case, the Hamiltonian is a measure of the total energy of the system and is defined classically as $H=\frac{{p}^{2}}{2m}+V\left(\stackrel{\to }{r}\right)$ . In many classical systems it's a conserved quantity. $H$ also lets you calculate classical equations of motion in terms of position and momentum. One big jump to quantum mechanics is that position and momentum are linked, so knowing 'everything' about the position (the wavefunction $\psi \left(\stackrel{\to }{r}\right)\right)$ at one point in time tells you 'everything' about momentum and evolution. In classical mechanics, that's not enough information, you must know both a particle's position and momentum to predict its future motion.

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