Understanding the Significance of Schrödinger Equation in Quantum Mechanics

[Sterj]

First: Hello all.

Theres a very important equation in non-relativistic quantum mechanic called schrödinger equation. I'm sure you know it. I searched for it in this forum but couldn't find what i want to find. I attached a file with this equation.

And now my questions:

I know the wave function psi(x,t) but psi(x,y,z,t)? Someone know an internet site that explains how it comes to this psi(x,y,z,t)? Or can someone explain me that?

Whats the meaning of this driangle? I guess it's called nabla.

Rest is clear if i am right.

I know the basics of physics (compton effect, photoelectric effect, ...) but what do I need indeed to derive the schrödinger equation?

Thanks for your answers.

 

[Galileo]

$$\nabla^2 = \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)$$

$\Psi(x,y,z,t)=\Psi(\vec r, t)$. It's the three-dimensional wavefunction.
Actually, this is what you normally start with and use $\Psi(x,t)$ after you've justified the use of the one-dimensional version.

You can't really derive the Schrödinger equation (in any case not with classical physics). It's a new fundamental law. Although it seems it can be made very plausible by using the Planck and the de Broglie relations.
From my viewpoint, you just have to assume it. It's experimental verification will prove its validity.


EDIT: overlooked the absence of the square in the operator...

 

[arildno]

The symbol $$\nabla$$ is commonly used for the gradient operator, i.e:
$$\nabla=\vec{i}\frac{\partial}{\partial{x}}+\vec{j}\frac{\partial}{\partial{y}}+\vec{k}\frac{\partial}{\partial{z}}$$
it's also called the "del"-operator.

The operator $$\nabla^{2}$$ is called the Laplace operator, is defined as:
$$\nabla^{2}=\frac{\partial^{2}}{\partial{x}^{2}}+\frac{\partial^{2}}{\partial{y}^{2}}+\frac{\partial^{2}}{\partial{z}^{2}}$$

The Schrödinger equation you've posted, is merely the 3-D generalization of the "ordinary" 1-D version you've seen before.

 

[QuantumDefect]

can someone explain how shroedinger created it also?

 

[Galileo]

http://fangio.magnet.fsu.edu/~vlad/pr100/100yrs/html/chap/fs2_14001.htm
I haven't read it yet, but it's Schrödinger's paper (or part of it) on a new quantum theory which he published in 1926. Should be a good read.



EDIT: Oh my god! I have medals! :!) :!) :shy: :shy:

 

[Sterj]

There are many signs like this "6" in the schrödinger equation. Has this "6" something to do with the differential equations?

Can somebody tell me, what I have to know to understand the schrödinger equation?

for example: I have to know the complex numbers, the wave functions, the de Brogli relations, and what else?

And, I searched for the "del-operator" and "Laplace-operator" but couldn't find a good internet site that explains how to derive this operators.

@Galileo: You said: You can't really derive the Schrödinger equation (in any case not with classical physics).
And can we derive this equation with the wave theories and some know how in quantum mechanic?

 

[Galileo]

There are many signs like this "6" in the schrödinger equation. Has this "6" something to do with the differential equations?

Do you mean the $\partial$ sign?
It's a partial differential. For example $\frac{\partial}{\partial t}$ is an 'instruction' to take the derivative with respect to t.
These derivatives are what makes the Schrödinger equation a differential equation ofcourse.

Can somebody tell me, what I have to know to understand the schrödinger equation?

That depends on what you mean by 'understand'. If you wish to know how this equation came to be. You need to inderstand langrangrian and hamiltonian mechanics, some results from optics and you'll be able to get your answer from the link I posted before.

And, I searched for the "del-operator" and "Laplace-operator" but couldn't find a good internet site that explains how to derive this operators.

These operators are not derived in any way. They are defined.
It makes for a nice and powerful notation.
These operators are frequently used in vector calculus.

Maybe Mathworld can help further: http://mathworld.wolfram.com/Laplacian.html

@Galileo: You said: You can't really derive the Schrödinger equation (in any case not with classical physics).
And can we derive this equation with the wave theories and some know how in quantum mechanic?

I don't know whether the Schrödinger equation is a logical consequence of the observed phenomena and other physical considerations.
It's a postulate of the theory and the fact is that the postulates of QM lead to predictions that are in agreement with experiment.
Mind you, that we can never fully prove or derive any physical theory, like the way you prove a mathematical theorem.

 

[Sterj]

@Galileo: Thanks a lot.

 

[jtbell]

In the USA, physics students usually study the Schrödinger equation for the first time in the second year of college or university, in a course which is usually called something like "Introduction to Modern Physics." There are many textbooks for this kind of course, for example Beiser's "Concepts of Modern Physics" which I taught out of for many years, and Taylor, Zafiratos & Dubson's "Modern Physics for Scientists and Engineers" which I'm now using.

To understand the Schrödinger equation in the sense of being able to solve it for specific situations, or at least understand how it predicts the properties of (say) the hydrogen atom, you need to know integral and differential calculus including partial derivatives, basic concepts of differential equations, and complex variables.

 

[Sterj]

I can do almost everything with the complex values (for example: r(cos(a)+i*sin(a))=r*e(i*a) its called the identity of Euler (in German), calculating with integrals isn't a problem).

But I never had (partial) differential equations. Someone knows an internet site that explains it (to a non-physic-student)?

 

[dextercioby]

I can do almost everything with the complex values (for example: r(cos(a)+i*sin(a))=r*e(i*a) its called the identity of Euler (in German), calculating with integrals isn't a problem).

But I never had (partial) differential equations. Someone knows an internet site that explains it (to a non-physic-student)?

Euler is a name,it comes from the famous mathematician Leonhard Euler and this is his name in any language.By the way,he was a German speaking Swiss.

PDE-s on the net??That's a tough one...U should be looking for books.If u can't buy'em,go to the nearest library and search for them.
Are u finding that difficult to move your butt and go to the library...?Being sedentary damages your health...

 

[theFuture]

I think it's important to try not to get caught up in all the details of the math when you're first introduced to the equation. There are a few problems you can work that require very little math but are important physical examples. This will give you insight into how the physics work as well as ease you into the mathematics. The 1-dimensional infinite square well, finite square well and simple harmonic oscillator are all good places to begin. From there, you'll feel much more comfortable with how things work and you can begin working with 3-d and other more complicated problems.

 

[Sterj]

@dextercioby
Euler is a name,it comes from the famous mathematician Leonhard Euler and this is his name in any language.By the way, he was a German speaking Swiss

I know that Leonard Euler was a Swisss mathematician living in the same city like Immanuel Kant, ... but here (in Switzerland) we call that
r*cis(a)=r*e^(i*a) the eulerian identity. I dont't refer the eulerian identity on the person, I refer it on this equation. (This equation e^(pi*i)+1=0 was (or is) the nicest equation in mathematics, cause of the basic values 1,0,e,pi,i).