Schrodinger equation

[Johnny5]

The Schrodinger equation is complex because the time evolution operator is complex. And the time evolution operator is complex because it must be unitary, which guarantees probability conservation. In other words, if the i is missing from the Schrodinger equation, its solutions will decay instead of wave.

 

Can you explain this a bit more.

 

time evolution operator??

 

[math] i\hbar \frac{\partial}{\partial t} [/math]

 

?

 

You say this leads to solutions which wave, and that if the i wasn't there then the solutions would decay.

 

Can you prove that mathematically, in as simple a manner as you know how?

 

Thanks

[Tom Mattson]

time evolution operator??

 

[math] i\hbar \frac{\partial}{\partial t} [/math]

 

?

 

That's not the time evolution operator. It's the total energy operator. The time evolution operator is:

 

[math]

U(t' date='t_0)=exp(\frac{-iH(t-t_0)}{\hbar})

[/math']

 

You say this leads to solutions which wave, and that if the i wasn't there then the solutions would decay.

 

Can you prove that mathematically, in as simple a manner as you know how?

 

You should be able to do it yourself. Write down the Schrodinger equation without the i, separate variables, and look at what happens to the temporal part.

[Johnny5]

That's not the time evolution operator. It's the total energy operator. The time evolution operator is:

 

[math]

U(t' date='t_0)=exp(\frac{-iH(t-t_0)}{\hbar})

[/math']

 

 

 

You should be able to do it yourself. Write down the Schrodinger equation without the i, separate variables, and look at what happens to the temporal part.

 

Ok will do.

 

Thanks