Problem 4.27 Quantum Mechanics

 (%i1) assume(hbar>0);

 (%i2) b:matrix(  [3*%i],  [4] );

 (%i3) c:conjugate(transpose(b));

 (%i4) c.b;

 (%i5) b:b/5;

 (%i6) Sx:matrix(  [0,hbar/2],  [hbar/2,0] );

 (%i7) Sy: matrix(  [0,-%i*hbar/2],  [%i*hbar/2,0] );

 (%i8) Sz: matrix(  [hbar/2,0],  [0,-hbar/2] );

Determine the expectation value of Sx, Sy, and Sz

 (%i9) expSx:conjugate(transpose(b)).Sx.b;

 (%i10) expSz:conjugate(transpose(b)).Sz.b;

 (%i11) expSy:conjugate(transpose(b)).Sy.b;

Determine the expectation of Sx^2, Sy^2, and Sz^2

 (%i12) expSx2:conjugate(transpose(b)).Sx.Sx.b;

 (%i13) expSy2:conjugate(transpose(b)).Sy.Sy.b;

 (%i14) expSz2:conjugate(transpose(b)).Sz.Sz.b;

 (%i15) sigmax:sqrt(expSx2-expSx^2);

 (%i16) sigmay:sqrt(expSy2-expSy^2);

 (%i17) sigmaz:sqrt(expSz2-expSz^2);

Test on the generalized uncertainty principle
applied to angular momentum: Eq. 4.100 page 161
and the cyclic permutations of Eq. 4.100
Note: I have rearranged to compare with zero
Griffiths Quantum Mechanics

 (%i18) test1:sigmax*sigmay-hbar/2*abs(expSz);

 (%i19) test2:sigmay*sigmaz-hbar/2*abs(expSx);

 (%i20) test1:sigmaz*sigmax-hbar/2*abs(expSy);

 (%i21) eigenvalues(Sx);

 (%i22) eigenvalues(Sy);

 (%i23) eigenvalues(Sz);

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