Schrodinger's equation

[sriram]

Guy's i have a doubt in Schrodinger's equation!

 

Im a beginner, so, anyone plz help me to understand the full equation clearly.

 

Thanks in advance.

[Klaynos]

http://en.wikipedia.org/wiki/Schrodinger_equation

 

Which bit don't you get?

 

How good is your maths education?

 

(why is this in chem?)

 

 

||well that's 2 edits wonder if I can think of anything else to add....

[Perturbation]

Schrodinger quantum mechanics has the time dependence in two things: the state-vector and consequently the state operator. There is no time dependance in the operators that represent dynamical variables. The Schrodinger equation expresses the time dependence of state-vectors as a differential equation, i.e. as a dynamical equation.

 

Start with your state-vector [imath]|\psi (0)\rangle[/imath] defined at initial time t=0 and perform a unitary evolution

 

[math]|\psi (t)\rangle =e^{-iHt/\hbar}|\psi (0)\rangle[/math]

 

Differentiate with respect to time

 

[math]\frac{\partial}{\partial t}|\psi (t)\rangle =\frac{-iH}{\hbar}e^{-iHt/\hbar}|\psi (0)\rangle =\frac{-iH}{\hbar}|\psi (t)\rangle[/math]

 

Putting this in the usual form gives

 

[math]H|\psi (t)\rangle =i\hbar\frac{\partial}{\partial t}|\psi (t)\rangle[/math]

 

This is the Schrodinger equation.

 

For a non-relativistic particle the Hamiltonian H is

 

[imath] H=\frac{P^2}{2M}+V [/imath]

 

And in coordinate representation the wave-function [math]\psi (\vec{x}, t)=\langle\vec{x} |\psi\rangle[/math] obeys

 

[math]\left(\frac{-\hbar^2\nabla^2}{2M}+V(\vec{x}, t)\right) \psi (\vec{x}, t)=i\hbar\frac{\partial}{\partial t}\psi (\vec{x}, t)[/math]

 

The momentum operator [imath]\vec{P}[/imath] is [imath]-i\hbar\vec{\nabla}[/imath] in coordinate representation.

 

For a relativistic particle we have the Dirac equation

 

[math]\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi (\vec{x}, t)=0[/math]

 

Heinsenberg QM differs in that the state-vectors and state operators have no explicit time depedence; instead the time dependence is in the dynamical operators, such as the Hamiltonian or Momenta.