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Find the possible total energies (Quantum Physics)

  1. Feb 5, 2017 #1
    1. The problem statement, all variables and given/known data
    I'm doing problems for practice in quantum physics.
    Consider two particles of the mass m in one dimension with coordinates being denoted by x and they are
    connected by a spring with spring constant k. Suppose that the total momentum of the system is p.
    Find all possible total energies for the following cases :
    (1)two particles are different (2)two particles are identical fermions.

    2. The attempt at a solution
    (1) I try to guess the answer...
    Total energy is the sum of potential and kinetic energy. Now our particles have the same mass and they are one-dimensional. Moreover, they are non-identical. Now potential energy is based on spring constant K therefore V=1/2*K*x^2 . Now considering the harmonic oscillator in classic sense total energy E= T +V
    Therefore E = P^2/2m + 1/2*mω^2*x^2.

    considering energy from quantum mechanical point of view, we know P= -iℏ d/dx =p' and x=x' hamiltonian becomes, H= 1/2 p'^2/2m + 1/2*mω'^2*x^2
    now considering the particles time independent
    H'ψ(x) = Eψ(x)
    the eigenvalues of this Hamiltonian is based on En = (n+ 1/2)ℏω where ground state has non-zero energy.

    (2) I have no idea how to start this problem.

    Thank you for your help.
     
  2. jcsd
  3. Feb 5, 2017 #2

    PeroK

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    I'm not sure at all how to interpret this problem. My first thought was that you have two particles in a harmonic oscillator, but I suspect that is not what is intended.

    I'm not sure how you can have two fermions connected by a spring. How could you attach a fermion to a spring?

    Perhaps what is meant is simply to calculate the potential for a classical system of this type and then translate that to a quantum potential - based on particle repulsion and attraction, rather than a spring! - and solve the resulting Schrodinger equation.

    In any case, your solution to 1), which appears to be simply a solution to a single particle SHO, cannot be correct.
     
    Last edited: Feb 5, 2017
  4. Feb 5, 2017 #3
    Thanks a lot.
    I upload the original problem file.

    I will try to solve the problem (1).
     

    Attached Files:

  5. Feb 5, 2017 #4

    PeroK

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    Okay, so the question setter does believe you can attach fermions and bosons to a spring. On the face of it, the natural length of the spring should be relevant, but I think you'll just have to do the maths and see what happens.
     
  6. Feb 5, 2017 #5

    TSny

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    You have a two-particle system. So, you can consider the wavefunction to be a function of two position coordinates: ψ(x1, x2).

    Do you know how to do a coordinate transformation which separates out the center-of-mass motion and the motion relative to the center of mass (with a reduced mass μ)?
     
  7. Feb 8, 2017 #6
    I tried to solve this problem for two days.
    But I still do not know how to solve it...
     
  8. Feb 8, 2017 #7

    TSny

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    I suggest that you change variables from ##x_1## and ##x_2## to ##X_c## and ##x##, where ##X_c## is the coordinate of the center of mass and ##x = x_2 - x_1##. This is a standard method for reducing the two-body problem to two independent one-body problems.

    Hopefully you've seen this before. See the first 3 or 4 pages here: http://physics.oregonstate.edu/~corinne/COURSES/ph426/notes2.pdf
     
  9. Feb 8, 2017 #8

    Thank you so much !!
     
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