I'd like to get some help and pointers on how to derive a wave function. Since I'm not really good at physics I'm just going to use brute force, if anyone spot a mistake please please point it out.(adsbygoogle = window.adsbygoogle || []).push({});

If I look at an electron in an well with infinite potential walls. The well has the dimensions -a<x<a

[itex]\displaystyle \frac{\hbar^2}{2m} \frac{\Psi d^2}{dx^2} = E \Psi [/itex]

[itex]\displaystyle k^2 = \frac{2mE}{\hbar^2}[/itex]

Gives

[itex]\displaystyle \frac{\Psi d^2}{dx^2} = -k^2 \Psi[/itex]

Solutions to diff equation

[itex]\displaystyle \Psi(x) = A \cos kx + B \sin kx [/itex]

Boundarys [itex]\displaystyle \Psi(\pm a) = 0 \rightarrow B = 0[/itex] and [itex]\displaystyle A = 0[/itex] or [itex]\displaystyle \cos ka = 0[/itex]

[itex]\displaystyle k=\frac{\pi(2n - 1)}{2a}[/itex]

Now

[itex]\displaystyle \Psi(x) = A \cos \frac{\pi(2n - 1)}{2a} x [/itex]

I also know that the integral from -a to a of the square should be equal to one.

[itex]\displaystyle \int^{-a}_a \left|\Psi(x) \right|^2\,dx = 1[/itex]

Can now be written like this

[itex]\displaystyle \int^{-a}_a A^2 \cos^2 \left( \frac{\pi(2n - 1)}{2a} x \right) \,dx = 1[/itex]

Some trigonometry

[itex]\displaystyle A^2 \int^{-a}_a \frac{1}{2} \,dx + A^2 \int^{-a}_a \cos \left( \frac{\pi(2n - 1)}{a} x \right) \,dx = 1[/itex]

[itex]\displaystyle A^2 a + A^2 \frac{\pi(2n - 1)}{a} \sin \left( \frac{\pi(2n - 1)}{a}(+a) \right) - A^2 \frac{\pi(2n - 1)}{a} \sin \left( \frac{\pi(2n - 1)}{a} (-a) \right) = 1[/itex]

Since the sin parts becomes zero.

[itex]\displaystyle A = \frac{1}{\sqrt{a}}[/itex]

Finally

[itex]\displaystyle \Psi(x) = \frac{1}{\sqrt{a}} \cos \frac{\pi(2n - 1)}{2a} x [/itex]

I take it this solution is in the interval -a < x < a, in all other points the function is zero. If this is somewhat correct, is there a more elegant way of doing this, or are there some standard solutions that can be used to find a particular solution?

Edit, how do I derive a wave function if a wall is at a certain potential, not infinite? For instance an electron in the third state in a 5 eV potential wall. I know that the function is exponential declining. But how do I determine where the nodes are and how high the wave function is at x = a?

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# Deriving a wave function

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