kmm

Gold Member

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## Main Question or Discussion Point

I worked out the expectation values of the components of a 1/2 spin particle. However, I'm confused about Griffiths notation for the x and y components.

For the x component I got, ## \left< S_x \right> = \frac \hbar 2 (b^*a+a^*b)## which is correct, but Griffiths equates this to ## \hbar~Re(ab^*) ##.

For the y component I got, ## \left< S_y \right> = \frac \hbar 2 i (ab^*-a^*b)##, and Griffiths equates this to ## - \hbar~Im(ab^*)##.

All I know is that Re and Im refer to the real and imaginary parts of a complex number. Since we only have a factor of ##i## for the y component of spin, it makes sense why we use Im here, but I don't understand how ##Re(ab^*) ## or ## Im(ab^*)## absorbed the factor of 1/2. I also don't understand why for both, we have ## ab^*## only inside the indices, when for the x component we have the factor ##(b^*a+a^*b)## and for the y component after factoring out the minus sign, we have the factor ##(a^*b-ab^*)##. Thanks for any help with this.

For the x component I got, ## \left< S_x \right> = \frac \hbar 2 (b^*a+a^*b)## which is correct, but Griffiths equates this to ## \hbar~Re(ab^*) ##.

For the y component I got, ## \left< S_y \right> = \frac \hbar 2 i (ab^*-a^*b)##, and Griffiths equates this to ## - \hbar~Im(ab^*)##.

All I know is that Re and Im refer to the real and imaginary parts of a complex number. Since we only have a factor of ##i## for the y component of spin, it makes sense why we use Im here, but I don't understand how ##Re(ab^*) ## or ## Im(ab^*)## absorbed the factor of 1/2. I also don't understand why for both, we have ## ab^*## only inside the indices, when for the x component we have the factor ##(b^*a+a^*b)## and for the y component after factoring out the minus sign, we have the factor ##(a^*b-ab^*)##. Thanks for any help with this.