# Schrodinger eqn. and its relativistic generalisations

#### masudr

I've suddenly run into a problem. This has probably arisen from the fact that I've yet to have (and may never have) formal teaching on the relativistic generalisation of QM.

I see the Schrodinger equation proper as

$$\hat{H}| \psi (t) \rangle = i\hbar \frac{d}{dt} | \psi (t) \rangle$$

Is this valid in the relativistic case? I guess it must be because wherever I have seen relativistic generalisations, they tend to be relativistic generalisations of the Hamiltonian of the single particle classical Hamiltonian $p^2/2m.$ And as it happens we can recast the above equation into some covariant form (is this coincedence or is it meant to happen?)

But then I later realised that the Dirac equation is more the equation of motion for the spinor field, i.e. in QED, we use the Dirac (for electron spinor field) + EM (for photon field) + interaction (electron-photon) Lagrangian.

In any of this, is the Schrodinger equation proper ever replaced by something else? Or is all we do is find Hamiltonians that describe our relevant particles? And does this apply to particles, or the associated field, or both?

Last edited:

#### dextercioby

Homework Helper
I've suddenly run into a problem. This has probably arisen from the fact that I've yet to have (and may never have) formal teaching on the relativistic generalisation of QM.

I see the Schrodinger equation proper as

$$\hat{H}| \psi (t) \rangle = i\hbar \frac{d}{dt} | \psi (t) \rangle$$

Is this valid in the relativistic case?
Ys, it is.

I guess it must be because wherever I have seen relativistic generalisations, they tend to be relativistic generalisations of the Hamiltonian of the single particle classical Hamiltonian $p^2/2m.$ And as it happens we can recast the above equation into some covariant form (is this coincedence or is it meant to happen?)
Of course it's not a coincidence, if the eqn is Lorentz invariant, then the Lorentz scalars must be made visible, i.e. using Lorentz space-time indices.

But then I later realised that the Dirac equation is more the equation of motion for the spinor field, i.e. in QED, we use the Dirac (for electron spinor field) + EM (for photon field) + interaction (electron-photon) Lagrangian.
Do you miss a "than" btw "more" and "the equation" ? If so, then you're wrong. The (IN)HOMOGENOUS Dirac eqn always describes the dynamics of the quantized Dirac field.

In any of this, is the Schrodinger equation proper ever replaced by something else? [/QUOTE

Well, in QFT, the SE properly describes the time evolution of state vectors, just like in the Galilei-invariant QM. So if you're worried about time-evolution of quantum (multi/uni)particle states, then you'd be worring about the equation posted by you.

Or is all we do is find Hamiltonians that describe our relevant particles? And does this apply to particles, or the associated field, or both?
Canonical quantization, even for free quantum field theories, is a mathematically complicated problem. It should be rigorously done using axiomatic field theory, either the Wightman formulation, or the Haag-Araki one. I'd say the path-integral approach to QFT is the least troublesome method, that is of course if you don't really inquire what a path-integral is from the mathematician's point de vue.

Daniel.

#### Demystifier

2018 Award
And does this apply to particles, or the associated field, or both?
In my opinion, this is one of the most important not yet satisfactorily solved questions in physics. For more details see
http://arxiv.org/abs/quant-ph/0609163
especially Secs. VII-IX.

#### reilly

In my opinion, this is one of the most important not yet satisfactorily solved questions in physics. For more details see
http://arxiv.org/abs/quant-ph/0609163
especially Secs. VII-IX.
The cited paper's author doesn't get it. He's attacking a strawman. Anyone who's practiced QM knows full well that the idea of wave-particle duality is a useful metaphor, nothing less, nothing more. It, I think, makes the notion of and direct evidence for electron diffraction easier for many to grasp -- knowing of course that such a description is ultimately a fiction. This is worth a big deal of concern? (One of my QM professors, J.H. VanVleck, used to describe a beam of electrons as a flight of mosquitoes. Should he give up his Nobel Prize for being so simple minded as to equate inanimate objects with animate opjects? And, as I recall, he didn't even warn us that he was using figurative language.)

Regards,
Reilly Atkinson

#### quetzalcoatl9

The cited paper's author doesn't get it. He's attacking a strawman. Anyone who's practiced QM knows full well that the idea of wave-particle duality is a useful metaphor, nothing less, nothing more. It, I think, makes the notion of and direct evidence for electron diffraction easier for many to grasp -- knowing of course that such a description is ultimately a fiction. This is worth a big deal of concern? (One of my QM professors, J.H. VanVleck, used to describe a beam of electrons as a flight of mosquitoes. Should he give up his Nobel Prize for being so simple minded as to equate inanimate objects with animate opjects? And, as I recall, he didn't even warn us that he was using figurative language.)

Regards,
Reilly Atkinson

out of curiosity, did you read the paper? i thought it was interesting, if somewhat uninhibited.

your professor probably didn't want to confuse you.

#### reilly

out of curiosity, did you read the paper? i thought it was interesting, if somewhat uninhibited.

your professor probably didn't want to confuse you.
Yes, I read the paper. Or, more correctly, I read more than half, but decided that reading more was not of interest to me. And, I do believe that is my right, without any explanation. (As a jazz musician, I say, man, that's a bunch of jive, ain't makin' the changes.)

My professor was prone to jokes, used wonderful figurative language, and, rightfully so, assumed his student were sufficiently bright and sophisticated in the use of mathematics and of language to understand and benefit from his deviations from the straight and narrow. Generally speaking, most of us, even non-physicists can tell figurative language from straight and pragmatic expression. He was not in the slightest interested in any behavior that might demean his students.

Reilly Atkinson

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving