Problem 4.27 Quantum Mechanics

(%i1) assume(hbar>0);

Result

(%i2) b:matrix(
 [3*%i],
 [4]
);

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

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(%i4) c.b;

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(%i5) b:b/5;

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

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

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

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Determine the expectation value of Sx, Sy, and Sz

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

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

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

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

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

Result

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

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

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

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

Result

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

Result

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);

Result

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

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

Result

(%i21) eigenvalues(Sx);

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(%i22) eigenvalues(Sy);

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(%i23) eigenvalues(Sz);

Result

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