The Role of Eigenvectors and Eigenvalues in Quantum Mechanics

  • Thread starter MadRocketSci2
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In summary, the conversation discussed the use of Schrodinger's equation and wave mechanics in understanding quantum physics, particularly in relation to electrons and photons. The concept of particles ending up in stationary states and the role of operators in measurements were also mentioned. The idea of "collapse" in the "shut up and calculate" interpretation of QM was brought up, along with its similarities to the theory of relativity. Decoherence was proposed as a solution to some of the issues with QM.
  • #1
MadRocketSci2
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I am currently working on trying to understand some of the details in quantum physics.

So far, Schrodinger's equation and wave mechanics seems to provide (at least when imposing semiclassical E-M interactions with an E-M field) a decent mechanism for why electrons end up in stationary states represented by the eigenfunctions of the wave equation. (These are the ones for which the wavefunction is not accelerating, thus not disturbing the E-M field.)

I have looked through second quantization, but haven't seen any mechanism there for quantizing photons. It's sort of asserted in the axioms of setting it up.

I've seen in it stated in several places that when particles interact, they always end up in an eigenfunction of some given operator. In general, I don't see the mechanism which would cause this. First, it seems the operators are arbitrarily chosen.

In the case of electrons, if you ping a bound electron with another electron, the other electron will eventually end up in some sort of stationary state by radiation (and eventually, the ground state). But if you have some generic particle that doesn't have a mechanism for dissipating excess energy, it doesn't seem like there should be any reason that it ends up in a stationary state versus some arbitrary time-varying state.

PS - for arbitrary time-varying states, it seems like the eigenfunctions are just some arbitrary basis to decompose them into. What is special about the eigenfunctions?

I've got a model I'm playing with now, with imaginary 1d particle A that is bound in a harmonic oscillator, and imaginary 1d particle B, that starts at one end of the domain and travels as a wave packet. There is some potential between them, so they interact when they "collide", and you end up with a wave for B that either propagates through (associated with one relative state for A), or reflects, associated with another relative state for A. But A, it seems, has no preference as to where it ends up, whether eigenvalue or not.

You have to come up with some specification for B's state anyway, before you get A's state. And you can specify whatever you want for B, for greater or lesser portions of the amplitude. Am I supposed to be seeing large amounts of the amplitude associated with eigenvectors?
 
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  • #2
A stationary state is an eigenstate of the Hamiltonian.

In general, a system will not end up in a stationary state.

When a measurement corresponding to an operator is made on system, the system will "collapse" from whatever state it is into an eigenstate of the operator corresponding to the measurement.

If the measurement is of the system's energy, then the system will collapse into a stationary state.

This "collapse" of the wavefunction occurs in the "shut up and calculate" interpretation of QM. It's obviously bizarre, and is one of the problems that say MWI tries to fix.
 
  • #3
Hi Mad Rocket

Its got to do with how to encode the results of observations. Imagine you have some kind of observational apparatus that has n outcomes and you associate n real numbers with those outcomes. You can list them out and you immediately recognize its a vector. Trouble is go to another basis and the values change - what you want is a way of expressing it that does not depend on a particular basis. To do that QM replaces the basis vectors |bi> by operators |bi><bi| to give the hermitian operator sigma yi|bi><bi| where the yi are the n real numbers. In this way it does not depend on any particular basis - the outcomes are the eigenvalues of the operator and the states are the eigenvectors.

In fact you can develop all of QM this way by imposing the condition any Hermitian operator in principle represents some observation and applying Gleasons Theorem - but that is another story.

Thanks
Bill
 
  • #4
The main reason given is repeatability: the measurement gives the same result if repeated. That means the state remains in the subset of state space corresponding to the value of the first measurement.

The "value" is not just a number on a gauge, it's also all the knowledge that went into the setup of the experiment, ie the entire known state.
 
  • #5
atyy said:
This "collapse" of the wavefunction occurs in the "shut up and calculate" interpretation of QM. It's obviously bizarre, and is one of the problems that say MWI tries to fix.

I know this is off-topic, and I don't want to sidetrack this thread, but I would be interested to know what you think of this statement: "The idea that the speed of light is constant irrespective of the speed of the observer is obviously bizarre, but the theory of relativity basically says "shut up and calculate".
 
  • #6
Rap said:
I know this is off-topic, and I don't want to sidetrack this thread, but I would be interested to know what you think of this statement: "The idea that the speed of light is constant irrespective of the speed of the observer is obviously bizarre, but the theory of relativity basically says "shut up and calculate".

It's not bizarre - it follows from the POR and Maxwell's equations. The thing with relativity is you are faced with an inconsistency - namely between locality with a finite propagation speed of effects and time as an absolute - both are heuristically very plausible but only one can be correct - it turns out to be locality that is correct.

Decoherence has solved many of the issues with QM IMHO, but I don't want this to be a discussion about that - it has be thrashed out in many threads on this forum already.

Thanks
Bill
 
  • #7
bhobba said:
It's not bizarre - it follows from the POR and Maxwell's equations. The thing with relativity is you are faced with an inconsistency - namely between locality with a finite propagation speed of effects and time as an absolute - both are heuristically very plausible but only one can be correct - it turns out to be locality that is correct.

Decoherence has solved many of the issues with QM IMHO, but I don't want this to be a discussion about that - it has be thrashed out in many threads on this forum already.

Thanks
Bill

Ok. I disagree, but I never posed the problem that particular way to one of your persuasion. Thanks for the answer.
 

FAQ: The Role of Eigenvectors and Eigenvalues in Quantum Mechanics

Why are eigenvectors/eigenvalues important in linear algebra?

Eigenvectors and eigenvalues are important concepts in linear algebra because they help us understand how linear transformations behave. Eigenvectors represent the directions in which a transformation only stretches or compresses, while eigenvalues represent the amount of stretch or compression along those directions. This information is crucial in many applications, such as principal component analysis, image processing, and differential equations.

How do eigenvectors and eigenvalues relate to each other?

Eigenvectors and eigenvalues are closely related in that an eigenvector always corresponds to a specific eigenvalue. In other words, an eigenvalue cannot exist without an associated eigenvector. This means that the eigenvalues and eigenvectors of a linear transformation are like two sides of the same coin and cannot be separated.

Why do we use eigenvectors and eigenvalues in data analysis?

Eigenvectors and eigenvalues are used in data analysis because they allow us to reduce the dimensionality of a dataset while preserving the most important information. By representing the data in terms of its eigenvectors and eigenvalues, we can identify the most significant patterns and relationships within the data. This is especially useful for large datasets where visualizing and interpreting the data may be a challenge.

What is the geometric interpretation of eigenvectors and eigenvalues?

The geometric interpretation of eigenvectors and eigenvalues is that they represent the directions and amounts of stretch or compression of a linear transformation. In other words, an eigenvector is a direction in which the transformation only scales the vector, and the corresponding eigenvalue is the amount of scaling along that direction. This can be visualized as the transformation of a circle or ellipse in the direction of the eigenvector.

How are eigenvectors and eigenvalues calculated?

Eigenvectors and eigenvalues can be calculated by solving a system of linear equations. In matrix form, the equation for finding the eigenvalues and eigenvectors of a square matrix A is (A - λI)x = 0, where λ is the eigenvalue and x is the corresponding eigenvector. This equation is known as the characteristic equation, and its solutions give the eigenvalues and eigenvectors of A.

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