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quick_shot · 2005-03-23 20:07:35

Could some one please explain what schrodingers equations are? I'm aware of the Uncertainty principle but don’t know what Schrödinger’s equations actually mean. What is there actual use????

jason_c · 2005-03-24 00:31:42

The total energy of a particle with mass m, momentum p, and potential energy V can be written as

$\frac {p^2}{2m} + V = E$

But in quantum mechanics p is no longer the momentum, but becomes the momentum operator. I won't go into the math details of this operator but just keep in mind that the p^2/2m term only resembles kinetic energy. So when the above equation is used in quantum mechanics, it only resembles total energy E. The momentum operator OPERATES on a wavefunction Psi (greek letter Psi). The wavefunction only represents the probability distribution of finding a particle with a certain energy. Schrodingers Equation calls the left side of the above equation H, and says

$H \Psi = E \Psi$

Since Psi is a probability distribution is why I say that the H in Schrodingers equation only resembles energy. H is composed of operators, so guess what. H is an operator. H operates on a wavefunction Psi, and the result is the energy E which is what you measure in the lab.

So to answer your question, schrodingers equation is used to find the energy of, for example, an electon. By the way, theres not really a connection per se between schrodingers equation and the uncertainty principle, at least not that I'm aware of.

Last edited by jason_c (2005-03-24 00:32:20)

Martin · 2005-03-24 05:19:18

quick_shot wrote:
Could some one please explain what schrodingers equations are? I'm aware of the Uncertainty principle but don’t know what Schrödinger’s equations actually mean. What is there actual use????

In the early part of the 20th Century, it became apparent that electromagnetic radiation—which had long been considered to be a purely wave phenomenon—often (specifically, when interacting with matter) displayed characteristics of particles. Along came Louis De Broglie, who suggested that since electromagnetic waves could sometimes be described as particles (“quanta”), might not particles sometimes be characterized by “particle waves?” Guided by some of Einstein’s work with the photoelectric effect (specifically, the relation between the energy carried by a quanta of electromagnetic radiation and the frequency of the electromagnetic radiation), De Broglie postulated some properties of “pilot waves” (the term he used to refer to the “particle waves”—waves that somehow might represent particles). Erwin Schrodinger picked up on the idea and developed it further. The equation he developed that explains the behavior of the “particle waves” is known as the Schrodinger Equation. The particle waves described by the Schrodinger Equation are interpreted as being related to the probability of finding the particle at a given point in space.

Martin · 2005-03-24 15:26:07

jason_c wrote:
By the way, theres not really a connection per se between schrodingers equation and the uncertainty principle, at least not that I'm aware of.

Indeed there is a connection. (There must be, if Schrodinger’s Equation is to be a valid tool (which it is) for describing the behavior of matter in the quantum world!)

Matrix Mechanics is a formulation of Quantum Mechanics created by Werner Heisenberg; it involves representing the dynamical properties of matter (such as position, momentum, and energy) by matrices, and incorporating the quantum aspects via his Uncertainty Principle.

Though Matrix Mechanics does not include concepts such as the wave function (Schrodinger’s term for the “particle waves”), the two approaches (Schrodinger’s and Heisenberg’s) were proven to be mathematically equivalent by David Hilbert. Thus, the solutions to Schrodinger’s Equation inherently incorporate the Uncertainty Principle (i.e., the solutions to Schrodinger’s Equation contain all of the information that the Uncertainty Principle allows us to know about the associated particles).

Martin · 2005-03-29 07:03:12

jason_c wrote:
...
...
...Psi is a probability distribution...

It's not the wave function ψ, but its amplitude squared that's a probability distribution.

jason_c · 2005-03-29 21:30:27

I figured, based on the phrasing of the question, that the explanation was borderlining too complicated. So I didn't want to complicate things further by saying the Psi is only an amplitude.

Martin · 2005-03-30 03:48:26

jason_c wrote:
I figured, based on the phrasing of the question, that the explanation was borderlining too complicated.

Frankly, I think that the "explanation" you offered was rather confusing. (I could be wrong, but I'm willing to bet that quick_shot didn't find it helpful.)

So I didn't want to complicate things further by saying the Psi is only an amplitude.

ψ is a function, not a amplitude. (I think that you understand that, but wanted to make sure that fact was clear.) Nevertheless, I doubt that, by that point in your explanation, the distinction between |ψ| and |ψ| $^2$ would have added to quick_shot's confusion.

daphysicist · 2006-02-27 22:37:05

Well lets see...

$\frac{h^2}{8 \pi^2 m} \nabla ^2 \Psi + V \Psi = \frac{h}{2 \pi i} \frac{\par \Psi}{\par t}$

Last edited by daphysicist (2006-03-01 20:24:43)

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#1 2005-03-23 20:07:35

Schrodingers Equations

#2 2005-03-24 00:31:42

Re: Schrodingers Equations

#3 2005-03-24 05:19:18

Re: Schrodingers Equations

quick_shot wrote:

#4 2005-03-24 15:26:07

Re: Schrodingers Equations

jason_c wrote:

#5 2005-03-29 07:03:12

Re: Schrodingers Equations

jason_c wrote:

#6 2005-03-29 21:30:27

Re: Schrodingers Equations

#7 2005-03-30 03:48:26

Re: Schrodingers Equations

jason_c wrote:

#8 2006-02-27 22:37:05

Re: Schrodingers Equations

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