partial differential equation (schrodinger)

[CPL.Luke]

alright so I have a textbook on quantum mechanics that I'm going through and it comes to the schrodinger equation and tells me to use a wavefunction equal to

 

[math]f(x)exp(-iEt/hbar)[/math]

 

I insert that into the schrodinger equation wich is

 

i* hbar *(partial derivative of the wavefunction with respect to t) = -hbar^2 /2m *(the second partial derivative of the wave function with respect to x)

 

 

it said that this simplified to

 

P.S.

sorry about latex Ill play with it and see if I can figure out how to work it

 

d^2 f(x)/dx^2 + (k^2)f(x)=0

 

this simplification shouldn't work this way because the wave function is of the form X(x)Y(y) right?

[ydoaPs]

[math]i{\hbar}\frac{\partial}{{\partial}t}\Psi(r,t)=-\frac{{\hbar}^2}{2m}\Delta{\Psi}(r,t)[/math]

[CPL.Luke]

thankyou for the latex

[Tom Mattson]

d^2 f(x)/dx^2 + (k^2)f(x)=0

 

this simplification shouldn't work this way

 

Yes' date=' it should work that way. In this case you will have [imath']k^2=\frac{2mE}{\hbar^2}[/imath].

 

because the wave function is of the form X(x)Y(y) right?

 

You mean X(x)T(t), right?

[CPL.Luke]

yeah, I meant one function multiplied by anouther of another variable

 

but thankyou for the answer

 

 

oh yeah in order to do a second order partial differential equation do you need to have 1 of the functions given?

[Dave]

No, you can assume the answer is X(t)T(t), and then obtain 2 ODEs from this, allowing you to get solutions fairly easily.