# Solution for schrodinger equation

In the potential well example we are considering the potential in the well to be zero and infinite outside the boundary, does this mean that the electron can move freely such that there is no opposition or restoring energy acting on it.

And also the probability of finding the electron is given by the product of wave equation and conjugate of wave equation modulus. How does this work?

Any kind of help would be appreciated.

Thank you.

malawi_glenn
Homework Helper
For the first one, you must think QM, the particle obey the shrodinger eq. The particle does not 'move freely', the walls are imposing bondary conditions on the eigenfunctions which will give you that only certain states are allowed (sinusodial ones). If the potential was 0 over the whole interwall (x from minus infinity to plus infinity) then we have a true free particle.

For the second, one uses:
$$\int _a^b \psi ^*(x) \psi (x) dx$$

To find the probablity that the particle, described by the normalised wavefunction psi, is between x = a and x = b.

jtbell
Mentor
In the potential well example we are considering the potential in the well to be zero and infinite outside the boundary, does this mean that the electron can move freely such that there is no opposition or restoring energy acting on it.

The classical analog is a particle that has zero net force acting on it inside the box, except at the walls. When it hits the walls, it rebounds perfectly elastically and instantaneously.

And also the probability of finding the electron is given by the product of wave equation and conjugate of wave equation modulus. How does this work?

Consider the wave function for the ground state of the "particle in a box" with walls at x = 0 and x = L:

$$\Psi(x,t) = \sqrt{\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right) \exp \left( - \frac{i \pi^2 \hbar t}{2mL^2} \right)$$

Then the probability distribution is

$$P(x,t) = \Psi^*(x,t) \Psi(x,t)$$

$$P(x,t) = \left[ \sqrt{\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right) \exp \left( + \frac{i \pi^2 \hbar t}{2mL^2} \right) \right] \left[ \sqrt{\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right) \exp \left ( - \frac{i \pi^2 \hbar t}{2mL^2} \right) \right]$$

$$P(x,t) = \frac{2}{L} \sin^2 \left( \frac{\pi x}{L} \right)$$

which doesn't actually depend on t in this case, although in general P does depend on both x and t.

Thank u, now i am able to understand the concept of finding an electron using the product of the wave fct and it's conjugate. I am referring to Interactive quantum mechanics by seigmund brandt, hans dieter dahmen and tilo stroh. I am a beginner do u suggest this book is good. As the concepts are mentioned but the equations are very direct I'm not able to get to know where did he start from.

Thank you once again for helping me.

Is there any text that will use mat lab programming to make the concepts clear and also teach the quantum physics theory.

Hey siddharth this one is cool...these applets are interesting and the visual part of it makes things clear.....Thanks bro, if any more sites for understanding things better whether books or any applications which will help in making concepts clear please let me know...