Quantum Physics, Schrödinger Equation...

Hey there.

I'm just finishing my Calculus course and I'm finally able to understand integration and diferantiation and even have fun with it. So I went to the library at school and picked up a Quantum Physics book. I was puzzled. At first, I understood the elementary particle physics of it, but then when I got to the Schrödinger Equation I was left completely puzzled. I think I can't understand what "momentum" is and stuff like that. I've studied this kind of think only in theory, a long time ago.

Can someone refresh my memory? What the heck is momentum? And what's up with the Schrödinger equation?

 

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Momentum is simply mass times velocity.

Take a look at this link:Schrodinger

 

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Hi Truthseeker

I remember when I first saw the Schrodinger equation and it was strange to think that such a weird equation could do anything at all! It turned out that it is very good at predicting what tiny (quantum) particles do in certain environments. In fact it even works for massive (classical) particles.

The Schrodinger equation may at first be overwhelming but it is in fact quite simple to understand. If you followed John Conellan's link then you should have the equation. The simplest is the Time-Independent Schrodinger Equation which does not involve time. The one that involves time is the Time-dependent equation which involves partial derivatives because the wavefunction becomes a function of two variables. Anyway, let me start with the TISE...

The bit out the front (the h-bar^2/2m bit) is merely a constant like 5x^2 where 5 is the constant. You should be able to recall h-bar off the top of your head (1.055x10^-34J.s) and m is the mass of the particle (in the case of an electron this would be 9.11x10^-31kg).

The next bit (the d^2Phi/dx^2) is a the second derivative of the wavefunction. Much like d/dx is a first derivative and d^2/dx^2 is a second derivative where x is a function.

The Phi is a function just like any other and is a special function which is unique to particles. It could be a wave function like Asin^iwt for instance.

As you move along you have a U(x)Y(x) bit. This bit deals with the energy of the particle expressed as a product of potential energy and its wavefunction. This part is needed when you are dealing with potential energy problems like wells and barriers (and indeed quantum tunnelling). The U(x) is a function of energy with position so we can deal with position of the particle in these circumstances.

This all equals the total energy function E(x)Y(x). Which is fairly straight-forward.

Now that you have a firm grasp on calculus (I have assumed you have done partial derivatives) you can solve this equation for various scenarios. Lets have a look at one...

Lets start with the TISE. Now stick the particle we are dealing with inside a potential well. Sitting inside the potential of the well is 0 (like sitting at the bottom of a real well. Potential energy is zero right? Mechanics) Anyway, the Schrodinger equation is simplified to (h-bar^2/2m)d^2Y(x)/dx^2 + 0 = EY(x).

d^2Y(x)/dx^2 = (-2mE/h-bar^2)Y(x) just rearranging.

Let k = sqrt(-2mE/h-bar^2) so

d^2Y(x)/dx^2 = k^2 Y(x)

This is a second order ODE and will have two linearly independent solutions namely

y(t) = Asinkx and y(t) = Bsinkx

This tells us that inside the well the particle has a sinusoidal wave pattern!! Isn't that interesting! Experiment will show that this is true! Classical mechanics does not say that when we are sitting at the bottom of a water well that we are oscillating - but Quantum mechanics says we do. From this many many interesting things pop up.

Does the particle it exist in the region where the potential of the wall of the well is greater than the energy of the particle? If you were Newton you would say no because E < U. But what the Schrodinger equation say?

Like before we solve for U(x) = U so the equation becomes...

h-bar^2/2m d^2Y(x)/dx^2 + UY(x) = EY(x)

d^2Y(x)/dx^2 = (2m(U - E)/h-bar^2)Y(x)

Let a = sqrt(2m(U - E)/h-bar^2) so...

d^2Y(x)/dx^2 = a^2Y(x)

Note how a is postive whereas before k was negative!!!!

This gives a different set of solutions by solving, namely...

y(x) = Ce^ax and y(x) = De^-ax

This says that the particle does exist outside the well!! And that it has an exponential form (unlike before when it was sinusoidal). Wow! The Schrodinger equation is telling us that the particle can exist in places where Classical mechanics forbids it.

But one thing must be noted. The Schrodinger equation cannot be derived from simpler equations. It is the simplest equation and works only because it agrees with experimental measurement.

 

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And because of that, it is only one interpretation of the quantum theory that works.

 

The Schrödinger equation can be derived from a classical Hamiltonian and the correspondence principle. Luckily it is not the axiom of all quantum mechanics, or else we wouldn't have spin, hyperfine splitting, or quantum field theory.

 

Yeah... now I see that I need partial derivatives. I was fooling around with my math book today and I found that there were two topics that weren't covered in the course: partial derivatives and double integrals. I think that's why quantum physics looked so confusing with all thos "(x,y)" things flowing around the pages...

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Oh! That was a constant? It looked like a variable. Oh well... I guess I don't know all teh constants, eh?

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How do you get the phi from taking the derivative of h-bar^2/2m? Where does it come from?

I don't know the latest function, but I understand what you are saying...

Yikes!

Yeah... it is always like that, eh?

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Yeah well... I guess I stop right here...
I need some more advanced math. I still have an exam on the end of the month. I'm comfortable enough to be bored with the "simple" math (which I considered to be overwhelming 4 months ago)...

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Oh... ok. That's enlightening...

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Just as a side note...

Don't you guys think that we absolutely need a source code that actually takes into account math symbols? I mean... we can always use < sup > to make this: e^x.... but that about all that we can do.

It's kinda irritating to having to write "integral" instead of just using a symbol or writing dx/dy instead of the "dy" being actually below the "dx".

Oh well... that's just me. Looking at those equations in the webish htmlish crapy format give me a headache...

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If there is such a code, can someone please give me some referance? Thanks.

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Type & int ; with no spaces to get an integral sign: &int;

You can adjust the size of the integral sign to make it more apparent. This is only useful for writing indefinite integrals though, as putting in limits of integration is still a pain.

Here is a list of HTML codes for various symbols, which can make things a little more presentable. The administrators of ************* have incorporated a TeX editor into the board software which enables people to present any equation properly. I think they are using the same software as this board, so I'm sure it would be possible to incorporate such an equation editor here, although it would require persuading the adminstrators to take the effort.

 

When you write *************, do you mean physicsforum ?

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Hopefully they will be incorporated, eh?

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Yeah well... which forums were those "*"s?

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If you spell phys*csf*r*ms properly it gets censored out. It must be a rude word around here.

 

are we really not allowed to mention ************* here at sciforums? if so, this is a bad form of censorship...

 

Apparently so, since I see the ************* showing up in everyone's threads (who apparently spell it correctly). That is quite a shitty form of censorship. Somewhat ironically, my explicitive has <i>not</i> been censored.

 

hey, Im just sayin.....

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Haha someone must be jealous of Physicsforum s !!!!

 

We are not differentiating -(h-bar)&sup2; / 2m. This is just a number - when &Psi; is describing an electron for instance this number is just -6.1088x10&sup-39;

&Psi; is just a symbol which we use to denote a matter wave.

The Time-independent Schrodinger equation...

(-(h-bar)&sup2; / 2m) d&sup2;&Psi;(x)/dx&sup2; + U(x)&Psi; = E(x)&Psi;

...is just like any ordinary equation like

a/3 d&sup2;x/dx&sup2; + x&sup3; = x&sup2;

But &Psi; is a function of position and time. Thus it is a multivariable function. When you differentiate a multivariable function you have to differentiate with respect to one of the functions. In the Time-Independent case &Psi; is a function of one variable - position - only, so we can differentiate normally. If you want to know what &Psi; is, its simplest form is sinusoidal which is

&Psi;(x) = &radic;(2/L) sin&sup2;(n&pi;x/L)

Where L is the width of the potential well.

If you insert this into the Time-independent Schrodinger equation then you will find that this is a solution.

The solution for the Time-dependent Schrodinger equation...

(-(h-bar)&sup2; / 2m) &part;&sup2;&Psi;(x,t)/&part;x&sup2; + U(x)&Psi;(x,t) = (ih-bar) &part;&Psi;(x,t)/&part;t

...is different because it involves a sum of linearly independent solutions. But I cant remember it off the top of my head.

But in the TDSE &Psi; is a function of two variables - &Psi;(x,t). When you differentiate it you have to keep time constant and differentiate with respect to x. Like x&sup2;t&sup3; differentiated w.r.t. x is like treated t as a constant. So it would be 2xt&sup3;. Differentiating w.r.t. t would be 3x&sup2;t&sup2;.

The point is that &Psi; is just a function of position and/or time. When you have the TDSE it is a multivariable function and the derivatives become partial derivatives. For the TISE the function is just of position so we take normal derivatives. &Psi; can be any function but it is generally Gaussian (a sum of sines and cosines - this is from Fourier Series) because this fits nicely with the Uncertainty principle.

 

When you solve the Schrodinger equation to find psi (I apologize for not using your fancy-pants HTML symbols) what you are finding is an eigenfunction of the Hamiltonian, which is just a measure of the amount of energy a system has. That wave function is in a sense an information carrier; if you want to know the average value of any classical quantity that depends on position and momentum, you can find it using the wave function. The wave function itself is not a measureable thing, it's just something that happens to work so we go with it.

In more sophisticated formulations the wave function is replaced with a state vector, which is probably a better way to think about it, but it's essentially the same thing only formulated as operators on a physical Hilbert space of normed vectors (if you didn't understand that don't worry about it, if you work at it enough you will).

The Schrodinger equation is mathematically just a second order partial differential equation of (for the one particle case) between two and four variables, which is separable in some cases, which allows you to turn it into differential equations in one and one to three variables, which are easier to solve.

The actual solution to the time-dependent equation involves a factor of exp[-i H t/hbar], so you're exponentiating a differential operator, but for the sake of time-independent potentials, this factor is just exp [-i E t/hbar], so not much of a problem.

As one of my favorite professors enjoys to say, from confusion comes enlightenment.

 

From all I know Schrodinger equation cannot be derived from something more basic.All books I've read agree on this,Schrodinger only used an analogy with the situation existent in optics for the wavefunction,still the equation itself is 'invented',all that count being that it's interpretation agree with all observed facts.Can you give a reference for your assertion?

 

Yes, I'd like to see how it is derived from the classical Hamiltonian and correspondence principle.

 

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My brain just melted

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