Home » Ask & Discuss » Physics. » Modern Physics « Back to Discussion

Modern Physics

Blazing goIITian

Joined:	20 Jul 2008
Post:	610

19 Feb 2010 13:39:28 IST

0 People liked this

396

Schrodinger's solution

None

I have a doubt related to solution of Schrodinger's eqn.

Solution of Schrodinger's eqn. is given by -----

$\Psi(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}$

But the general solution of wave eqn. is

y = A sin(kx-wt)

And this even satisfies the eqn....Then, what was the necessity of depicting the solution as complex phase factor.

Share this article on:

Comments (5)

Yasin

Cool goIITian

Joined: 13 Jul 2008

Posts: 70

20 Feb 2010 15:07:13 IST

Like 0 people liked this

Hi Pratyush! Well the general wave equation that ur mentioning is the general equation for sine-cosine wave functions with A being amplitude.ANY wave can be defined as f(x,y,z,t)=S , where x, y, z and t are the coordinates of space-time and S is the solution space. For non-3dimensonal situations, we don't have all the space coordinate variables. Hence, u cn think of the schrodinger wave function as a subset of all wave fucntions and as u said, since the simple harmonic sine-cosine wave fits into the schrodinger eqn as well, it is a subset of the schrodinger general eqn.

Protyush Sahu

Blazing goIITian

Joined: 20 Jul 2008

Posts: 610

21 Feb 2010 10:32:35 IST

Like 0 people liked this

But if u look at the complex solution of Schrodinger's equation, it also satisfies the general

equation of 1-D wave which is given by ---

(partial^2psi)/(partialx^2)=1/(v^2)(partial^2psi)/(partialt^2).

So, both the solutions

$\Psi(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}$ (given by Schrodinger)

and

y=A Sin(kx-wt)

satisfy all the wave equation(1-D,2-D,3-D) . But Schrodinger is credited of finding the solution as complex phase factor. And this is also used as solution of matter wave.............Why ???

Yasin

Cool goIITian

Joined: 13 Jul 2008

Posts: 70

21 Feb 2010 13:55:43 IST

Like 1 people liked this

Bcoz matter waves, and wave functions that define the permitted quantum states available for electrons(or more generally, any particle that obey th Pauli eclusion principle, called fermions) in an atom (or more generally in any 'trapped potential state' like an electron is trapped to an atom) are not all neat sone-cosine waves. They are complicated functions that give solutions of the permitted quantum states of wave-particles in a system. Also, the credit goes to schrodinger because his equations provide th 'more general result' , inclusive of, as u said, matter waves etc. I think in the same way as relativity explains all of newtonian mechanics. Equations of relativity can account for both classical and modern physics mechanics but that does not necessarily mean we can use the newtonian mechanics to explain all situations, more precisely near-speed-of-light situations.

Sushobhan Sen

Blazing goIITian

Joined: 1 Jul 2009

Posts: 1416

6 Jun 2010 01:42:38 IST

Like 0 people liked this

Because the solution to the eq. is a function that described probability of finding a wave, and that is something we cannot measure. If we cannot measure it (ie, it is not an observable) it need not be real. Hence, the more general complex solution is used. Note that in maths, they should actually teach you the complex solution first and then the sine-cosine solution, as the former is the general solution and the latter a special case. We can disregard the complex part if and only if we are sure that the answer does not have a complex component. In this case, we are not sure and so we don't ignore it.

edison

Forum Expert

Joined: 19 Oct 2006

Posts: 7537

28 Jun 2010 14:59:19 IST

Like 0 people liked this

The wave function is complex

Actually this is nature is forced upon us so tat the equation satisfies all 4 basic conditions.

Moreover the equation contains an 'i' because it relates a first time derivative to a 2nd space derivative. This is due, in turn, to the fact that the Schrodinger equation is based on the energy equation which relates the first power of total energy to the second power of momentum.

The fact that wave functions are complex functions should not be considered a weak point of the QM. Actually it is desirable feature because it makes it immediately apparent that we should not attempt to give to wave functions a physical existence in the same sense that water waves have a physical existence. The reason is that complex quantity cannot be measured by actual physical instrument. The "real" world (using the term in its nonmathematical sense) is the world of "real" quantities (using the term in its mathematical sense).

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team