Re-arranging the Schrodinger equation

[vorcil]

I just have a small question,

In my book it says that the Schrödinger equation,

$$i\hbar\frac{\partial\Psi}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

rearranged is,

$$\frac{\partial\Psi}{\partial t} = \frac{i\hbar}{2m}\frac{\partial\Psi ^2 psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$

how does the complex number, move over, and in the numerator? instead of the denominatior?

I can see how $$A\hbar = B\hbar ^2 becomes A = B \hbar$$

but I don't understand how

$$A i = B + V\Psi becomes A = iB - i V\hbar$$

could someone please explain to me the mathematical rules behind rearranging complex numbers in equations,

or give me some links that explain it, (in simple terms) please :P

 

[Matterwave]

$$\frac{1}{i}=-i$$

This is one of the properties of imaginary numbers. I don't exactly recall a proof for this...hopefully someone else can answer your question in more detail.

 

[vorcil]

Do you know what I would search for If i wanted to understand how to manipulate i?

I tried the wikipedia log of complex numbers, but it is jungle of crap that is too hard to understand
tl;dr

 

[JaWiB]

$$\frac{1}{i}=-i$$

This is one of the properties of imaginary numbers. I don't exactly recall a proof for this...hopefully someone else can answer your question in more detail.

Like this?
$$\frac{1}{i} = \frac{1}{i}\frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$$

 

[vorcil]

Like this?
$$\frac{1}{i} = \frac{1}{i}\frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$$

JaWiB

how do i do this

$$A i = B + V\Psi to A = iB - i V\Psi$$

 

[JaWiB]

I don't think what you have is correct.
$$i\hbar\frac{\partial\Psi}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$
If you multiply both sides by $$i/\hbar$$, you get
$$-\frac{\partial\Psi}{\partial t} = i\frac{\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2} + \frac{i}{\hbar}V\Psi$$
or
$$\frac{\partial\Psi}{\partial t} = -i\frac{\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$

 

[vorcil]

mmmm well that's closer then to what I had,

thank you i'll ask my tutors tomorrow

 

[Cyosis]

You have the Schrödinger equation wrong. It should be:

$$i\hbar\frac{\partial\Psi}{\partial t} =- \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

 

[vorcil]

You have the Schrödinger equation wrong. It should be:

$$i\hbar\frac{\partial\Psi}{\partial t} =- \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

so Jawib's way does work!??

 

[vorcil]

I need to take some math papers...
I'm not as good as the rest of the physics majors at mathematics,

I've only done 1 math paper and 5 physics ones

I didn't even think to multiply both sides by i/h, like Jawib said :(

study study study

 

[Gregg]

I need to take some math papers...
I'm not as good as the rest of the physics majors at mathematics,

I've only done 1 math paper and 5 physics ones

I didn't even think to multiply both sides by i/h, like Jawib said :(

study study study

If you had kept at it, you would have got it eventually. You gave up. If you give up a lot that is a problem. Half-hearted studying won't get you anywhere.