Quantum interaction questions

[DanielFB]

eigenfunctions are values of operators, so i hear. the subshells of atoms contain neutrons or protons and when the atom is excited some particles jump out to higher states... is this correct to say?

 

in particle creation, so i hear, a photon is absorbed into a system. and when the particles annihilate a photon is released, right? but the two particles that are created have certain properties meaning that one is just like the other going backwards in time. so what we have now is a situation that involves a particle being created, emitting a photon, and then going back in time. that is my understanding.

 

semiempirical equations, eigenfunctions, excited states, neutrons, nucleons, particles, atoms, and so on. this is physics language.

 

light impinges upon certain molecules and the same reaction in return except this time with radiation. the molecules 'see' and gradually have learned to control the motion of atoms. And whether it likes it can make particles jump from state to state but of course, this is not clear. this is madness. but sanity is an equation.

[ajb]

eigenfunctions are values of operators,

 

Let [math]A[/math] be a linear operator on some vector space [math]V[/math] over complex numbers. (don't worry about if it is a topological vector space or anything like that).

 

An eigenfunction of the operator [math]A[/math] is an element [math]f \in V[/math] such that

 

[math]Af = af[/math]

 

with [math]a\in C[/math].

 

In quantum mechanics, eigenfunctions correspond to states that have exactly the property that measuring the observable corresponding to [math]A[/math] gives the value [math]a[/math]. (which must of course be real, but I am not sure how much we want to push this right now).

 

 

 

so i hear. the subshells of atoms contain neutrons or protons and when the atom is excited some particles jump out to higher states... is this correct to say?

 

Probably, one is thinking of electrons in shells surrounding the nucleus, rather than the states of the neutrons and protons in the nucleus.

 

But yes, nucleons can change state giving rise to gamma rays.

 

in particle creation, so i hear, a photon is absorbed into a system. and when the particles annihilate a photon is released, right? but the two particles that are created have certain properties meaning that one is just like the other going backwards in time. so what we have now is a situation that involves a particle being created, emitting a photon, and then going back in time. that is my understanding.

 

This sounds like a interpretation of Feynman's sum over all histories. So, ok but I am not sure if I would take this interpretation too far. Without a it of quantum field theory, it is hard to say too much about particles and antiparticles.

[adapa]

Let [math]A[/math] be a linear operator on some vector space [math]V[/math] over complex numbers. (don't worry about if it is a topological vector space or anything like that).

 

An eigenfunction of the operator [math]A[/math] is an element [math]f \in V[/math] such that

 

[math]Af = af[/math]

 

with [math]a\in C[/math].

 

In quantum mechanics, eigenfunctions correspond to states that have exactly the property that measuring the observable corresponding to [math]A[/math] gives the value [math]a[/math]. (which must of course be real, but I am not sure how much we want to push this right now).

 

 

 

 

 

Probably, one is thinking of electrons in shells surrounding the nucleus, rather than the states of the neutrons and protons in the nucleus.

 

But yes, nucleons can change state giving rise to gamma rays.

 

 

 

This sounds like a interpretation of Feynman's sum over all histories. So, ok but I am not sure if I would take this interpretation too far. Without a it of quantum field theory, it is hard to say too much about particles and antiparticles.

 

Hi. It's been a couple of decades since I've done anything that is physics related and I am trying to re-activate that part of my brain so please pardon my ignorance if I ask any silly questions.

 

From what I remember, I understand that wavefunctions are represented as vectors in a mathematical manifold called Hilbert Space. I also understand that the wavefunctions are the sums of the products of its eigenstates and their respective eigenvalues, and that these eigenstates are represented as mutually orthogonal unit vectors in Hilbert Space. Is that correct?

 

For example:

 

[math]\left|\psi\right\rangle=\sum^{n}_{i=1}C_{i}\left|\phi_{i}\right\rangle[/math]

 

Where [math]\left|\phi_{i}\right\rangle[/math] represents each eigenstate and C[math]_{i}[/math] represents each eigenvalue on n-dimensional Hilbert Space.

 

So I understand (or at least think I do) that the system is in a state of superposition whenever more than one of the eigenvalues is non zero. I also believe that the wavefunction is considered collapsed when there is only one eigenstate with a non-zero eigenvalue. Is that correct? If not, please feel free to steer me in the right direction.

 

Thanks for answering:-)

[ajb]

I think what you said is about right.

 

A Hilbert space is a kind of topological vector space. You can have what are known as Hilbert manifolds, which are locally modelled on Hilbert spaces. But this is a different subject.

 

I do not think in general that it is true that you can write any vector as the sum of eigenvectors for some operator. I think this is a postulate of quantum mechanics. (I need to check this carefully).

 

However, you do have orthonormal basis.

 

There is the spectral theorem for compact operators. This states that for any compact self-adjoint operator there exists an orthonormal basis consisting of the eigenvectors.

Edited November 1, 2009 by ajb

[adapa]

I do not think in general that it is true that you can write any vector as the sum of eigenvectors for some operator. I think this is a postulate of quantum mechanics.

Thanks for answering.

I obviously need to improve my understanding of this part.

 

Also, I have a question about operators. I understand that the operator can be represented as a square matrix. Now is a self adjoint operator described as one where the operator is equal to the transpose of the complex conjugate of itself? Or is that a different concept entirely?

[ajb]

That is right for finite dimensional Hilbert spaces. It becomes more difficult for infinite dimensional Hilbert spaces, but thankfully in general our intuition about finite dimensions holds.

[adapa]

That is right for finite dimensional Hilbert spaces. It becomes more difficult for infinite dimensional Hilbert spaces, but thankfully in general our intuition about finite dimensions holds.

 

Thanks again for answering and please pardon my interrogation.

 

When I had the misconception about the wavefunction being a sum of the products of the eigenstates and their eigenvalues, would it have been more accurate to say that each state is a linear combination of the mutually orthogonal basis states?

 

In other words, when I said that [math]\left|\psi\right\rangle=\sum^{n}_{i=1}C_{i}\left|\phi_{i}\right\rangle[/math]

 

Would it have been more accurate if [math]\left|\psi\right\rangle[/math] is the state, and [math]C_{i}\left|\phi_{i}\right\rangle[/math] is the product of the [math]i^{th}[/math] basis state and its coefficient? Or is this also just as wrong?

 

Thanks again for answering. Pardon the rust.

Edited November 1, 2009 by adapa

[ajb]

The "C's" do not have to be eigenvalues. They measure the probability of that state being chosen in the wave function collapse.

[adapa]

The "C's" do not have to be eigenvalues. They measure the probability of that state being chosen in the wave function collapse.

Thanks. That makes perfect sense. I am going to try to find some reading on the subject now. It will probably take me a couple of days before I know enough to have any further questions. I sincerely appreciate your help.