Schrodinger equations?

Question

Hi, i'm having real trouble understanding the schrodinger equations (the time dependent and time-independent), can anyone explain simply what the difference is between the two, and what they are used for ?
Anyone that explains this as simply as possible, and without the use of too much maths will get 10 points.
Thanks, David.

Answer 1

The above answer is great, but to get at the heart of what they are used for: they are used to model the behavior of quantum mechanical (usually small) systems. The t-d eq. gives the system's evolution in space and time, whereas states of constant energy, which vary trivially with time, the spatial part is of greater interest--say, to find the electron density of a molecule.

Answer 2

Of the two, the time-dependent SE is the more fundamental. The time-independent SE is an application of it, when you want to find specific energy values.

The RHS of each of them is the same: It's the Hamiltonian operator (the energy operator) that acts on the wavefunction. In the case of a point particle in a potential well, the RHS is like:
RHS = (-(hbar)^2/(2m) Laplacian + V(x)) f(x,t)

(Laplacian = d2/dx2 in 1-dimension)

where f(x,t) is the wavefunction.

The LHS of the time-dependent equation is:
LHS = i*hbar*df/dt , so the full time-dependent SE is:
i*hbar*df/dt = (-(hbar)^2/(2m) Laplacian + V(x)) f(x,t)

In the special case that the wavefunction is a quantum state with a definite energy value E, it has to look like:
f(x,t) = g(x)*exp(-i*E*t/(hbar)) , so
df/dt = (-i*E/hbar)*g(x)*exp(-i*E*t/hbar)) , and so:
df/dt = - (i*E/hbar)*f(x,t)
When you plug this into the time-dependent LHS, you get:
LHS = (i*hbar)*(df/dt)
= i*hbar*(-i)*E/hbar * f(x,t)
= E*f(x,t)
So if you take this special-case LHS, and equate it to the RHS, you get:

E*f(x,t) = (-(hbar)^2/(2m) Laplacian + V(x)) f(x,t)

Sorry to use all these equations, but you ARE asking about Schrödinger's equations!

The point is that the time-independent equation is a special case of the time-dependent equation, which you use to discover the specific values of E that will work - only a few values will work. It applies only when you assume that the time behavior is a complex exponential.

It is part of the folklore of physics that the SE is associated with skiing:
- First, when the original idea of the particle's wavelength was presented by de Broglie at a conference, Peter Debye commented, "Well, if we're going to say that particles behave like waves, isn't the grown-up approach to find an equation for the waves?" And then he and everyone else went skiing - except Schrödinger, who stayed home and found the wave equation (the time-dependent one, of course).
- Second, when the time came to write up the paper to fully lay out wave equation, Schrödinger went off on a skiing vacation with a young lady (who? unknown - but not his wife!) and during certain periods of the day, worked on these important papers. In particular, he generated solutions to many of the energy-value questions we still quote today (based on the time-independent SE) very easily - because a famous textbook in mathematical physics, by Courant & Hilbert, had essentially solved all these problems already! (They were just doing math, of course; they hadn't realized that Schrödinger was making up a theory that would employ their techniques so profoundly.)

I hope this provides some perspective.

Answer 3

Time-dependent is basically an academic exercise. He used the time-dependent equation to show that the wave equation is actually time-independent.

If you make the normal assumptions, time-dependent simplifies to time-independent. Time-independent is the form usually used for practical analysis.

Answer 4

This argument, whether one assumes all its premises to be real, only money owed for deism. At this factor, i can only ask if this God's lifestyles is of any effect to me. Does this God communicate with mankind? CAN this God communicate with mankind? Intelligence is an fairly summary theory. How can we are saying that God actual has intelligence in this occasion? If God's intelligence only applications as a lawlike connection, then God has approximately as lots expertise of the universe as a thermometer has expertise with regard to the ambient temperature. removing justification from any form of communicate concerning expertise, by way of definition, strikes you out of the area of epistemology altogether. If God extremely applications this way, then we will not call his intelligence "intelligence", its only a effortless action-reaction nature of the universe!

Answer 5

phisi(x,y,z,t)=0

partial differential equation,
d^2 (x)/dx^2 +d^2(y)/dy^2+d^2(z)/dz^2-Ud^2 (V)/dt^2=0
three dimentional wave equation.
z=rsinx,z=rcosy