The state space E(k,j,m) can be considered a direct sum of the orthogonal subspaces E(j,m), where m varies in integral jumps from -j to +j and j takes on all the values actually found in the problem.  The dimension of each subspace is g(j).  (Here each vector in a subspace E(j,m) has the same j and m but a different index k.)

We can also consider E(k,j,m) to be the direct sum of all orthogonal subspaces E(k,j) spanned by vectors with the same k and j but different indices m.  The dimensions of these subspaces are 2j+1, and the subspaces are globally invariant under the action of all components of J.  Any component of J acting on a vector in E(k,j) yields another vector in E(k,j) .

What are the matrices of J_z, J₊, and J_- in E(k,j)?

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The matrix elements in E(k,j) therefore are

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The matrix elements are independent of the value of k, they depend only on j and m, they are "universal".  They do not depend on a particular physical system.

Examples:

• j=0:

  The subspaces E(k,j) are one dimensional, the only possible value for m is zero.  All matrices reduce to one element, which is zero.

• j=½:

  In the { |+½>, |-½> } basis we have

  ,

  ,

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• j=1:

  In the { |1>, |0>, |-1> } basis we have

  

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• For an arbitrary j we have

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Problems:

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with the center of the charge distribution at the origin of the coordinate system.  Here

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A nucleus with a quadrupole moment has a quantum mechanical charge density r_aJM(r)µ|y_aJM(r)|², which depends on the quantum numbers J and M.  The quadrupole moment and the energy of the nucleus in an electric field gradient therefore depend on J and M.

Consider a nucleus with a quadrupole moment and with angular momentum l=1.  Assume that the Hamiltonian of the nucleus may be written as

,

where L_u and L_v are the components of L along the two directions in the x-z plane which form angles of 45^o with the x- and z-axes and is a constant.

(a) Write the matrix of H in the { |1>, |0>, |-1> } basis.  What are the stationary states of the system and what are their energies?  Write these states as |E₁>, |E₂>, and |E₃>, in order of decreasing energy.

(b) At time t=0 the system is in the state  .  What is the state vector |y(t)> at time t? At time t L_z is measured?  What are the probabilities of the various possible results?

(c) Calculate the mean values <L_x>(t), <L_y>(t), and <L_z>(t).  What is the motion performed by the vector <L>?

(d) At time t L_z² is measured.  Do times exist when only one result is possible?  Assume that the measurement yielded h². What is the state of the system immediately after the measurement?  Indicate, without calculation, its subsequent evolution.

• Solution:

  (a) . .

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  in the { |1>, |0>, |-1> } basis.

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  Assume is positive. Then

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  (b) .

  

  

  

  

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  (c) .

  

  =0.

  

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  =0.

  or

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  (d)   .

  At and |y(t)> is an eigenvector of L_z² with eigenvalue h².

  At and |y(t)> is an eigenvector of L_z² with eigenvalue 0.

  After a measurement of h² the state of the system is the eigenstate of L_z² . This state evolves like |y(0)>.

Show that if any operator commutes with two of the components of an angular momentum operator, it commutes with the third.

Solution:

• Concepts:
  The commutation relations for the Cartesian components of any angular momentum operator
• Reasoning:
  Commutation relations are what defines a vector operator as a angular momentum operator.  We define angular momentum through [J_i,J_j] = ε_ijkiħJ_k.

• Details of the calculation:
  Let  i ≠ j,k   j ≠ k and let i, j, k be cyclic ( x, y, z  or  y, z, x  or  z, x, y ).
  Then  [J_i,J_j] = iħJ_k, J_iJ_j - J_jJ_i = iħJ_k.
  Let A be an operator.
  Assume [J_i,A] = 0,  [J_j,A] = 0.  Then [A, J_k] = (1/iħ)[A,J_iJ_j - J_jJ_i].
  iħ[A, J_k] = [A,J_iJ_j] - [A,J_jJ_i] = [A,J_i]J_j + J_i[A,J_j] - [A,J_j]J_i - J_j[A,J_i] = 0.

Consider the so-called spin Hamiltonian , for a system of spin 1. Show that the Hamiltonian in the S_z basis is .  Find the eigenvalues of this Hamiltonian.

• Solution:

  For a spin 1 system we choose as the basis vectors the eigenstates of S_z, { |1>, |0>, |-1> } with eigenvalues respectively.  In this basis the matrix for S_z is

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  The matrices for S_x and S_y are

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  Therefore

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  We have  .

  The eigenvalues and eigenvectors are:

  

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