Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: syhprum on 09/04/2013 12:06:34
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I understand a few of the symbols the reduced Plank one for instance which that has a numerical value but what is H and t and why is i (-1^.5) there for and what do the vertical bars mean.
What was Schrodinger trying to tell us ?
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It shows the development of the wavefunction of a physical system over time - and also possible forms that the wavefunction may take.
H is the Hamiltonian Operator - is a quantum mechanical operator (http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html) and contains information about the total energy ie both the kinetic and potential energy of the system from the Hamiltonian (a function of position, velocity and time),
Phi is the Wavefunction (http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wvfun.html)(continuous function of position and time) which includes all information about a particle
d/dt is the partial derivative with respect to time
i and h are constants
the vertical lines | and pointy brackets > are kets from bra-ket notation (http://en.wikipedia.org/wiki/Bra–ket_notation)
To actually explain the meat is beyond me - and you must have noticed that there are oodles of different schroedinger equations
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I understand a few of the symbols the reduced Plank one for instance which that has a numerical value but what is H and t and why is i (-1^.5) there for and what do the vertical bars mean.
What was Schrodinger tryind to tell us ?
To answer you comprehensively and correctly would require some years of university physics, so I can begin with...small pieces and obviously someone else will reply "but that's not exact because..." [:)]
Level 0.5: It says how to relate the (quantum) system's wavefunction, for example of a particle, to the forces applied to it. The operator H (Hamiltonian, almost always correspond to the particle's energy) acts on the wavefunction (w.f.) as a differential equation for it; the partial derivative of the w.f. with rispect to time is the definition of that operator; so that equatino essentialy says: "the particle energy is: ..."
I know that this doesn't explain much, but we are just at level 0.5.
Remember that the w.f. is the quantum way of writing the particle's properties: instead of position and velocity (classical case for a point mass) you have a function of x and of t, from which you can compute what you want: approximately trajectory (en exact one doesn't exist in QM), energies, probability of finding the particle in every region of space and so on.
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Many thanks for the links I think I understand about 1% now of what it is all about
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But did you appreciate the answers even without the links? [:)]
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Yes indeed the elucidation of the meaning of the symbols was helpful although I do not appreciate the purpose of i that you call a constant but which I assume to be the qurt of -1.
It was informative that the vertical bars were Kets but of course without the link I would have had no idea what a Ket was.
Of course one like me with only a low level of mathematical skill makes the naïve assumption that mathematical formulae a simply algorithm's into which one can insert numerical values and get a meaningful answer but I now appreciate this is somewhat more complex.
Browsing thru the links I came across something with which I was familiar although I did not know the name "Gaussian elimination" this I was taught at school c1942 one thing I was not taught was about "i" we were told that some quadratics could not be solved which of course was not true.
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It's hard to really appreciate what that "i" is doing there without a solid mathematical background, but it has to do with the fact that this equation is describing waves. Physicists knew well before quantum mechanics that you could represent waves using complex numbers [meaning those numbers involve the sqrt(-1)]. Complex numbers are not measurable, so in some sense, you can view this as a tool for book-keeping, where at the end of the day anything measurable consists of real values extracted from the complex valued wave.
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Wolfram Mathematica has some interactive demonstrations where you can adjust the parameters in the Schrodinger equation and see the results.
PS I well understand the usefulness of "i" in electrical calculations
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PS I well understand the usefulness of "I" in electrical calculations
Cool! It's doing much the same thing in the Schrodinger equation that is is in phasors in electrical calculations. The quantum weirdness of Schrodinger is that we can write down the mathematics for classical waves or currents in electrical circuits and know that they are real things that can be measured. We use phasors to make calculations easy, obviously current or sound waves aren't physically complex-valued.
In QM, it's less clear what the "wave" really is physically, since our measurements yield particles, not a continuous wave. This is why there's a lot of debate over how to interpret the complex-valued wave function that the Schrodinger equation is describing. http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics