http://upload.wikimedia.org/wikipedia/commons/d/d9/1-D_Box.png

http://upload.wikimedia.org/math/2/0/2/20276c56b7727f9948f8a7cacfc05d52.png
This is one of the formula for "Partilce in a 1-D box" obtained from Schrodinger's Equation.

Can someone please tell me what "x" and "L" and "psi" mean in this formula? What should I substitute for "x" in a problem using this formula?

And is n*pi*x/L in degrees or raidans?

Thanks a lot!

x is the variable labelling position, L is the width of the box, and psi is just a symbol commonly used to represent the wavefunction. n*pi*x/L is in radians, so the calculus comes out pleasantly.
Problems involving this most pleasant of wavefunctions will either have to do with finding the probability that the particle is in a certain region of the box, by taking the integral between a and b of psi* *x*psi, or integrating the same over all space to find the normalisation coefficient.

L is the length of the box. So for example, if the length of the box is 2cm, do I substitute 2cm for both "x" and "L"? Are "x" and "L" always the same thing?

http://upload.wikimedia.org/math/2/0/2/20276c56b7727f9948f8a7cacfc05d52.png
For this formula, the variable on the left (psi) is wave function, what does it mean? What actually have I calculated after using this formula? Say, if I got a psi value of 5.258, what does it mean?

You would only substitute 2cm for L. x is a variable, while L is a parameter. �psi sub n as we have here is an eigenstate of energy for this system, n being an integer as the wavefunction must be continuous, and therefore zero at x=0 and L. As I stated above, you will only use this by taking the integral over some region of the product of the complex conjugate of the wavefn and some operator acting on the wavefn. This will yield the expected value of the observable associated with the operator in the region over which the integral is taken. If the operator is unity, the result will be the probability of finding the particle in the region.

kingwinner:

psi(n) is the 1-D probability amplitude function for the n-th energy level in the box. The lowest possible energy is n=1, the next allowed energy is n=2 and so on.

L is the length of the box (say in metres). It could be 2 cm, and is a constant.

x is the position where you want to know the value of the probability amplitude. If you plug in a value between 0 and L for x, the value of the function psi(n) is the probability amplitude that the particle will be found at that location in the box.

If you take the probability amplitude and square it, then you have the probability density for finding the particle at location x in the box. Probability density is probability per unit length.

So, if you want to find the probability that the particle is found between locations x=a and x=b in the box, the answer is:

Probability = (integral from a to b) Psi(n)^2 dx

for given values of n, L, a and b.

i see...so x is position

http://upload.wikimedia.org/wikipedia/commons/d/d9/1-D_Box.png
Is x=0 always the left boundary of the box?


I haven't learned any integration in calculus yet.
If I were asked to find psi for x=1.5cm, can I substitute a single value for x (position), say x=1.5cm and find psi?

Also, is it ture that psi itself means nothing until you square it?

I have another question...

The 2nd formula for partilce in a 1D box is:
http://upload.wikimedia.org/math/c/f/2/cf246592053c9e29ba3a80e14ba220bc.png

But what does this E_n represent? Energy of a single electron? Energy of a mole of electrons? Or...?

The particle in a box has discrete values of energy that are available to it, because of the boundary conditions. The subscript n is the quantum number used to label the different states, and E_n is the energy of the particle in the n-th energy state.

Integrating is finding the area under a section of the curve, so if you were to try and find the probability that the particle was exactly at 1.5 cm, the answer would be zero, as although the height is non-zero, the width is. While you can find a value for psi at a certain position, it is by itself of no use.

Is x=0 always the left boundary of the box?

In this example, yes.

I haven't learned any integration in calculus yet.
If I were asked to find psi for x=1.5cm, can I substitute a single value for x (position), say x=1.5cm and find psi?

Yes. But that only gives you the probability amplitude at that position.

Also, is it ture that psi itself means nothing until you square it?

No. Psi is the probability amplitude. It is the thing which evolves in time according to the Schrodinger equation. It is only when you want to make measurements that you need to square psi.

The 2nd formula for partilce in a 1D box is: ...

But what does this E_n represent? Energy of a single electron? Energy of a mole of electrons? Or...?

The particle in the box has mass m. It could be an electron, in which case m is the electron mass. It could be a tennis ball, in which case m is the mass of the tennis ball.

E_n is technically the allowed energies of the system consisting of the particle and the box.

In this example, yes.



Yes. But that only gives you the probability amplitude at that position.



No. Psi is the probability amplitude. It is the thing which evolves in time according to the Schrodinger equation. It is only when you want to make measurements that you need to square psi.



The particle in the box has mass m. It could be an electron, in which case m is the electron mass. It could be a tennis ball, in which case m is the mass of the tennis ball.

E_n is technically the allowed energies of the system consisting of the particle and the box.
For example, given the length of the 1D box is 10 cm, find psi and the probability of an electron at x=5.0 cm and n=1.

psi=sqrt(2/10) sin(1*pi*5.0 / 10) =0.4472 (by the way, should the L and x be in cm or in m when substituting into the formula? Becuase whether it's cm or m does change the value of psi...)

psi^2 = 0.2 (does this mean that there is a 20% chance of finding the electron at the position x=5.0 cm?)

=====================================

What is probability amplitude? What does psi mean, physically?

=====================================

For example, if n=1, mass of electron =9.1x10^-31 kg, length of box=100 m, then
E = (1)(h^2) / 8(9.1x10^-31)(100 m) = 9.1x10^-7 J
Is this 9.1x10^-7 J the energy per electron, or is it the electron per mole of electrons?

For example, given the length of the 1D box is 10 cm, find psi and the probability of an electron at x=5.0 cm and n=1.

psi=sqrt(2/10) sin(1*pi*5.0 / 10) =0.4472 (by the way, should the L and x be in cm or in m when substituting into the formula? Becuase whether it's cm or m does change the value of psi...)

Regarding units: L and x can be in any units of length you like, as long as they are the same. Inside the sin function, the units will cancel out.

The units do have an effect in the square root factor outside the sin function, but to find a probability, you need to first square psi, which also squares the units, so that the units of psi squared are 1 over the length units you chose.

Then, to get an actual probability, you integrate over the length, which cancels out the length units and gives you a probability that is just a number.

psi^2 = 0.2 (does this mean that there is a 20% chance of finding the electron at the position x=5.0 cm?)

No. It means the probability density (probability per unit length) of finding the electron at position x=5 cm is 20%. The actual probability of finding the electron at a particular location is zero.

I already told you how to calculate the probability that the electron is between any two locations. You need to integrate.

What is probability amplitude? What does psi mean, physically?

Psi is the wavefunction. In general in quantum mechanics, psi can have complex values (i.e. real and imaginary parts).

Psi actually contains everything that can be known about the properties of the system. When you square psi, you lose some of the information content - especially if psi is complex.

For example, if n=1, mass of electron =9.1x10^-31 kg, length of box=100 m, then E = (1)(h^2) / 8(9.1x10^-31)(100 m) = 9.1x10^-7 J
Is this 9.1x10^-7 J the energy per electron, or is it the electron per mole of electrons?

The function gives the energies of one single particle in a box. The energy is the total energy of the system, which in this case means potential plus kinetic energy of the particle.