Solution for schrodinger equation

In summary, the potential well example shows that the electron can move freely within the boundaries of the well, but the probability of finding it is given by the product of the wave equation and the conjugate of the wave equation modulus.
  • #1
charanshah09
5
0
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.
 
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  • #2
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:
[tex] \int _a^b \psi ^*(x) \psi (x) dx [/tex]

To find the probablity that the particle, described by the normalised wavefunction psi, is between x = a and x = b.
 
  • #3
charanshah09 said:
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:

[tex]\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)[/tex]

Then the probability distribution is

[tex]P(x,t) = \Psi^*(x,t) \Psi(x,t)[/tex]

[tex]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] [/tex]

[tex]P(x,t) = \frac{2}{L} \sin^2 \left( \frac{\pi x}{L} \right)[/tex]

which doesn't actually depend on t in this case, although in general P does depend on both x and t.
 
  • #4
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.
 
  • #5
Is there any text that will use mat lab programming to make the concepts clear and also teach the quantum physics theory.
 
  • #7
also, these applets may help

http://phet.colorado.edu/new/simulations/index.php?cat=Quantum_Phenomena
 
  • #8
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...
 

1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is named after Austrian physicist Erwin Schrodinger, who developed it in 1926.

2. What does the Schrodinger equation solve for?

The Schrodinger equation solves for the wave function of a quantum system, which contains information about the probability of finding the system in different states. It can be used to calculate the position, momentum, and other physical properties of a quantum system.

3. What is a solution to the Schrodinger equation?

A solution to the Schrodinger equation is a mathematical function that satisfies the equation and represents the wave function of a quantum system. It can be written as a combination of basis states, which are eigenfunctions of the Hamiltonian operator.

4. How is the Schrodinger equation used in chemistry?

The Schrodinger equation is used in chemistry to understand the behavior of atoms and molecules at the quantum level. It can be used to calculate the electronic structure of atoms and molecules, and to predict their chemical and physical properties.

5. What are some applications of the Schrodinger equation?

The Schrodinger equation has many applications in physics, chemistry, and other fields. Some examples include understanding the behavior of particles in a magnetic field, predicting the properties of semiconductors, and studying the behavior of nuclei in a nuclear reactor.

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