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? i bar(h)(partial)/(partial t)Psi=H Psi I thought wave equation should be of the type (partial^2)/(partial^2t)u=c^2 nabla^2u It's the differenc

anraszbx 2022-11-05 Answered
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?
i t Ψ = H Ψ
I thought wave equation should be of the type
2 2 t u = c 2 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?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

artirw9f
Answered 2022-11-06 Author has 20 answers
You should not think of the Schrödinger equation as a true wave equation. In electricity and magnetism, the wave equation is typically written as
1 c 2 2 u t 2 = 2 u 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
i ψ t = 2 2 m 2 ψ 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 ( x ) ψ 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 ψ ( r ): 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 ψ ( r ), we make the demand that all solutions must be normalized such that | ψ ( r ) | 2 d r = 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 ψ ( r ) 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 ψ is a solution to a linear equation, a ψ is also a solution for some (complex) a. However, we say all such states are 'the same', and anyway we only accept normalized solutions ( | a ψ ( r ) | 2 d r = 1). We say that solutions like ψ, and more generally e i θ ψ, represent the same physical state.
3. Some special solutions ψ E are eigenstates of the right-hand-side of the time-dependent Schrödinger equation, and therefore they can be written as
2 2 m 2 ψ E x 2 = E ψ E
and it can be shown that these solutions have the particular time dependence ψ E ( r , t ) = ψ E ( r ) e i E t / . 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 ψ and ϕ are solutions, so is a ψ + b ϕ, 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: 1 2 ( ψ + e i θ ϕ ) are solutions for all angles θ, 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 ψ ( r , t ) is a solution to the Schrödinger's equation at position r and time t, then the probability of finding the particle in a specific region can be found by integrating | ψ 2 | around that region. For that reason, we identify | ψ | 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 | ψ ( r , t ) | 2 d r = 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
< A >= ψ ( r , t ) A ^ ψ ( r , t ) d r
where A ^ is the linear operator associated to the observable. In the position representation, the position operator is A ^ = x, and the momentum operator, p ^ = i / 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 = p 2 2 m + V ( r ) . 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 ψ ( r ) ) 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.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-05-10
Which formula for the de Broglie wavelength of an electron is correct?
So, I have my exams in physics in a week, and upon reviewing I was confused by the explanation of de Broglie wavelength of electrons in my book. Firstly, they stated that the equation was: λ = h p , where h is the Planck constant, and p is the momentum of the particle. Later, however, when talking about electron diffraction and finding the angles of the minima, the author gave the formula equivalent to that for light: λ = h c E . Now, what I don't understand is if it is simply a mistake made by the author, or whether a different formula have to be used for electron diffraction, as the two formulae are very clearly not equivalent. In the latter case, I don't understand why the formula would be different. I greatly appreciate the help, as the exams are really close, and I would like to make sure I get this right!
Edit: I was told that pictures of text are taking away from the readability of the posts, and thus they were removed. Essentially, the difference between the two cases are that in the first case, the proton did not have any significantly large kinetic energy, while in the second example, the kinetic energy was 400   M e V
asked 2022-08-11
To determine
Which one have shorter wavelength when both proton and electron travels with same speed.
asked 2022-05-18
Combination of de Broglie wavelength and mass–energy equivalence gone wrong?
I tried to combine the mass–energy equivalence for a particle with mass,
E = ( m c 2 ) 2 + ( p c ) 2 = ( m c 2 ) 2 + ( γ m v c ) 2
with de Broglie wavelength,
λ = h p = h γ m v .
I get this equation:
E = h c 2 λ v .
This does not seem right, since the equations suggest the energy increases as the speed slow down which is not the case. But I can't see what I did wrong, either. Can someone help me?
asked 2022-11-03
10. In de Broglie's equation, he shows the relationship between the wavelength of a light wave and its momentum. Which of the following is true about this relationship?
1. wavelength is directly proportional to the square of momentum
2. wavelength is inversely proportional to momentum
3. wavelength is directly proportional to momentum
4. wavelength is inversely proportional to the square of momentum
asked 2022-11-04
To determine
To Describe: A method to measure energy of a photon.
asked 2022-07-20
The speed of electron.
asked 2022-05-14
Is the relation c=νλ valid only for Electromagnetic waves?
What is the validity of the relation c = ν λ? More specifically, is this equation valid only for Electromagnetic waves?
I read this statement in a book, which says:
"de Broglie waves are not electromagnetic in nature, because they do not arise out of accelerated charged particle."
This seems correct, but arises a doubt in my mind.
Suppose I find out the wavelength of a matter wave (or de Broglie wave) using de Broglie's wave equation:
λ = h p
Now, can I use c = ν λ to find out the frequency of the wave?

New questions