Schrodinger equation

[Guest muralisankar]

can anyone give me a complete explanation of the Schrodinger equation and also give its derivation

or atleast tell me where on the web i can find it

[swansont]

AFAIK it wasn't derived. Erwin just decided to use a wave equation to describe things, and it worked.

[[Tycho?]]

http://www.google.com

http://www.wikipedia.com

[ydoaPs]

[math]i{\hbar}\frac{\partial}{{\partial}t}\psi(r,t)=\frac{{\hbar}^2}{2m}\Delta\psi(r,t)[/math] well, thats what my book called Quantum Mechanics says.

[Johnny5]

[math]i{\hbar}\frac{\partial}{{\partial}t}\psi(r,t)=\frac{{\hbar}^2}{2m}\Delta\psi(r,t)[/math'] well, thats what my book called Quantum Mechanics says. i (square root of negative one) times hbar times the first derivitive of the wave function = hbar squared over 2 times the mass times change in the wavefunction.

 

That's not right. It's more complicated it's this...

 

[math] -\frac{ \hbar^2}{2m} \nabla^2 \Psi + U \Psi = i \hbar \frac{\partial \Psi}{\partial t} [/math]

 

Where

 

[math] \Psi = \Psi (x,y,z,t) [/math]

 

Now, if we let the position vector be denoted as follows...

 

[math] \vec r = x \hat i + y \hat j + z \hat k [/math]

 

then we can write:

 

[math] \Psi = \Psi (x,y,z,t) = \Psi (r,t) [/math]

 

In the case where the wavefunction is constant in time, the lowercase psi is usually used, that is:

 

[math] \psi = \Psi (x,y,z) = \Psi ® [/math]

 

 

 

 

You have some object with inertial mass m in some potential U.

 

hbar is Planck's constant of nature divided by 2 pi.

 

i is the square root of negative one.

 

Psi is called the wavefunction.

 

The fundamental postulate of quantum mechanics is that the momentum of an object is related to an inertial wavelength in the following way:

 

[math] |\vec P| = \frac{\hbar}{\lambda} [/math]

 

From classical mechanics we have:

 

[math] |\vec P| = |m \vec v| = m |\vec v| [/math]

 

The quantity |v| is the speed of the center of inertia of the object in some particular inertial reference frame. Speed is a strictly positive quantity, and is defined as distance traveled in the frame, divided by time of travel (as measured by a clock at rest in the frame).

 

Now, there are two kinds of velocity associated with a quantum mechanical inertial wave. They are called the group velocity , and the phase velocity. Here is a somewhat more detailed discussion of group velocity

 

Suppose that we equate the classical expression for momentum with the quantum mechanical expression for momentum. Then we arrive at this formula here:

 

[math] m |\vec v| = \frac{\hbar}{\lambda} [/math]

 

Now, suppose that the object with inertial mass m is moving through an inertial reference frame with nonzero speed |v|. Then we can divide both sides of the equation above to get this:

 

[math] m = \frac{\hbar}{|\vec v| \lambda} [/math]

 

So there is a problem introduced, if we try to analyze an object in a reference frame in which it's speed is zero.

 

At any rate, the LHS of the formula above gives the inertia of the object in terms of it's speed in a frame, and it's quantum mechanical wavelength.

[Dave]

I should probably point out that the Schrodinger equation is often introduced for a wavefunction of only two variables; x and t. i.e. we only consider what happens in one-dimensional space.

 

In this case, the equation simplifies down to:

 

[math] -\frac{ \hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + U \psi = i \hbar \frac{\partial \psi}{\partial t} [/math]

[Johnny5]

I should probably point out that the Schrodinger equation is often introduced for a wavefunction of only two variables; x and t. i.e. we only consider what happens in one-dimensional space.

 

In this case' date=' the equation simplifies down to:

 

[math'] -\frac{ \hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + U \psi = i \hbar \frac{\partial \psi}{\partial t} [/math]

 

You know, this is never adequately explained in texts, this shift from three dimensional form, to one dimensional form or vice versa.

 

What is the correct way to visualize the wave?

 

Do you know?

[Dave]

I don't think I've got the adequate knowledge to answer that properly. The way I visualized it was to imagine a particle on a line, but that's probably wrong

[Johnny5]

The original poster wanted a complete explanation that's a very tall order. The three dimensional form should be used, and the spherical harmonics for a hydrogen atom should be derived. Can anyone here do that for him?

 

To Muralisankar: Do you know spherical coordinates?

[swansont]

Tycho already addressed that. For finding straightforward facts, there's Google or some other search engine.

[Johnny5]

Tycho already addressed that. For finding straightforward facts, there's Google or some other search engine.

 

Yes but deriving the spherical harmonics of a hydrogen atom isn't a straightfoward fact. If the person already knows spherical coordinates, they could follow the derivation. Otherwise, at least instruct them on what to go learn, and return when they know it.

 

Google isn't the answer to a question that complex. I was going to derive them for him. But right now I am working on several problems which you have given me on gravity.

[swansont]

Yes but deriving the spherical harmonics of a hydrogen atom isn't a straightfoward fact. If the person already knows spherical coordinates' date=' they could follow the derivation. Otherwise, at least instruct them on what to go learn, and return when they know it.

 

Google isn't the answer to a question that complex. I was going to derive them for him. But right now I am working on several problems which you have given me on gravity.[/quote']

 

First hit on a Google search.

 

Third hit

 

You can at least proceed from there, and ask more pertinent questions about the concepts, rather than just ask for a recitation of facts and a derivation.

[Johnny5]

First hit on a Google search.

 

Third hit

 

You can at least proceed from there' date=' and ask more pertinent questions about the concepts, rather than just ask for a recitation of facts and a derivation.[/quote']

 

Wow, solutions on the web, first hit. Yeah that's the one. Hmm.

 

Well let me ask you something related then.

 

What is psi*psi?

[ed84c]

You are (unless im very much mistaken) referring to the Wave Function.

 

As far i understand it, this is the most difficult concept to understand in physics, i tried the same as you are and burnt.

 

but try!

[Johnny5]

You are (unless im very much mistaken) referring to the Wave Function.

 

As far i understand it' date=' this is the most difficult concept to understand in physics, i tried the same as you are and burnt.

 

but try![/quote']

 

Actually, quantum mechanics is all about the wave function.

 

It's not the most difficult concept of physics, I realize that is somewhat debatable, but no matter.

 

There is a disagreement as to what is doing the waving.

 

Regards

[ed84c]

No, thats not quite what i meant.

 

I meant how to use it mathematically, i think. Ill find you the quote in the book if you want.

 

Im quite happy with the waving of things in QED.

[Johnny5]

No' date=' thats not quite what i meant.

 

I meant how to use it mathematically, i think. Ill find you the quote in the book if you want.

 

Im quite happy with the waving of things in QED.[/quote']

 

Waves in a material medium are fine by me, probability waves are not.

 

What did you mean?

[ed84c]

I read, That using the Wave Function in QM is nay impossible, and the numbers from using it are just as hard to interpret.

 

Maybe i mis read, as i say, if i find the quote in the book (in search of shrodingers cat, i think) ill quote it here.

[Johnny5]

I read' date=' That using the Wave Function in QM is nay impossible, and the numbers from using it are just as hard to interpret.

 

Maybe i mis read, as i say, if i find the quote in the book (in search of shrodingers cat, i think) ill quote it here.[/quote']

 

In the case of the hydrogen atom, the equation has a great deal of success. It's the interpretation of psi which is in question. Not the empirical data.

 

Regards

 

PS: Of course one could hope for something better one day, but right now we are stuck with it.

[□h=-16πT]

Wow' date=' solutions on the web, first hit. Yeah that's the one. Hmm.

 

Well let me ask you something related then.

 

What is psi*psi?[/quote']

 

The probability. The asterix denotes complex conjungation.

[Johnny5]

The probability. The asterix denotes complex conjungation.

 

Probability of what?

[swansont]

Waves in a material medium are fine by me' date=' probability waves are not.

[/quote']

 

Nature doesn't care whether it's OK with you or not. The behavior of the world around us is not limited by your (anyone else's) level of understanding.

 

And argument from incredulity doesn't cut much ice in science.

[Johnny5]

Nature doesn't care whether it's OK with you or not. The behavior of the world around us is not limited by your (anyone else's) level of understanding.

 

And argument from incredulity doesn't cut much ice in science.

 

Well maybe I should expand on why it's not ok with me, but right now I have my hands full in the other thread.

[Saint]

Nature doesn't care whether it's OK with you or not. The behavior of the world around us is not limited by your (anyone else's) level of understanding.

 

And argument from incredulity doesn't cut much ice in science.

 

Probablity waves don't occur in nature. They are mental constructs.

[Johnny5]

Probablity waves don't occur in nature. They are mental constructs.

 

 

Something is waving though. I am hoping it's the inertia.