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Hey!

My teacher gave me a very challenging problem on QM.

I've only had one introductiary course, but he said he wanted me to figure this problem out by using whatever resources, so now I turn to you guys ;-). The problem is the following:

London attraction forces can be thought of as a pure QM effect that can be calculated with the usage of 2 level systems that can feel each other due to transition dipole moments (this is also called exciton interaction).

If we couple two 2-level systems (in NMR experiments this is called spin coupling) then four different energy levels will be possible. (These levels were present before coupling, but then they were two by two identical). The four levels can be found by determining the eigenvalues of the following matrix:

http://img91.imageshack.us/img91/585/matrix2ej.gif [Broken]

where E is the energy of the excited state of the original 2-level system (the ground state energy is 0) and U is given by:

http://img168.imageshack.us/img168/734/u8yw.gif [Broken]

with u1 and u2 two transition dipole moments and r = the vector that connects u1 and u2.

a) Determine the eigenvalues of the matrix. Hint: think of it as two easier-to-calculate 2x2 matices that you can diagonalize much easier. Sketch the position of the energy levels in a energy diagram.

b)What is the new ground state energy?

c) The two dipole moments are both pointing in the x- direction and the first one is located in the origin. The other one is in the position (0,0,2). Distances are measured in angstroms. Now sketch the ground state energy as function of the position of the second dipole moment along the z-axis.

d) What is the force that both systems exert on each other if their distance is z?

e) Show that the force, in a good approximation, is proportional to U

Could anybody of you PLEASE PLEASE PLEASE help me with this problem?!?! I'd REALLY appreciate it very very much!!!!

My teacher gave me a very challenging problem on QM.

I've only had one introductiary course, but he said he wanted me to figure this problem out by using whatever resources, so now I turn to you guys ;-). The problem is the following:

London attraction forces can be thought of as a pure QM effect that can be calculated with the usage of 2 level systems that can feel each other due to transition dipole moments (this is also called exciton interaction).

If we couple two 2-level systems (in NMR experiments this is called spin coupling) then four different energy levels will be possible. (These levels were present before coupling, but then they were two by two identical). The four levels can be found by determining the eigenvalues of the following matrix:

http://img91.imageshack.us/img91/585/matrix2ej.gif [Broken]

where E is the energy of the excited state of the original 2-level system (the ground state energy is 0) and U is given by:

http://img168.imageshack.us/img168/734/u8yw.gif [Broken]

with u1 and u2 two transition dipole moments and r = the vector that connects u1 and u2.

a) Determine the eigenvalues of the matrix. Hint: think of it as two easier-to-calculate 2x2 matices that you can diagonalize much easier. Sketch the position of the energy levels in a energy diagram.

**I thought using the first and fourth row in one 2x2 matrix and the others in another one, but where to go from here?**b)What is the new ground state energy?

**I really don't know how I'd solve this problem..**c) The two dipole moments are both pointing in the x- direction and the first one is located in the origin. The other one is in the position (0,0,2). Distances are measured in angstroms. Now sketch the ground state energy as function of the position of the second dipole moment along the z-axis.

**I really don't know how I'd solve this problem too..**d) What is the force that both systems exert on each other if their distance is z?

**Perhaps I need to take the - derivative to the position of the energy??? (but what would the energy function be....?)**e) Show that the force, in a good approximation, is proportional to U

^{2}and thus proportional to 1/distance^{6}, just like the London force. Hint: Use the Taylor series.**Again: I really don't know how I'd solve this problem..**Could anybody of you PLEASE PLEASE PLEASE help me with this problem?!?! I'd REALLY appreciate it very very much!!!!

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