For the state calculate the expectation values and...

For the state |+), calculate the expectation values and uncertainties for measurements of $S_x$, $S_y$, and $S_z$ in order to verify hint: $\langle S_z \rangle = \frac{\hbar}{2}$ $\Delta S_z = 0$ using matrix $\langle S_x \rangle = 0$ $\Delta S_x \neq 0$ notation is easier $\langle S_y \rangle = 0$ $\Delta S_y \neq 0$ remember, |+z> state is $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ bra is complex conjugate transpose

Transcript

00:01 All right, so we have a spin state that i've written here.

00:06 So, kai is a, some normalization constant a, and then our values are two and three i in the sort of spin up and spin down states.

00:16 And we want to first find the normalization constant.

00:19 So to find that, we just basically multiply, we have the condition that kye dagger times kai is equal to one.

00:30 And so when we do that, we multiply this by a constant.

00:32 It's conjugate transpose.

00:34 We get a squared.

00:36 And the conjugate transposed, by the way, is going to be, you know, a times two minus three i, something like this.

00:47 So we compute the inner product.

00:48 We get two squared plus three squared.

00:50 And so all of this requires that a is one over the square to 13.

00:54 So that's our normalization concept.

00:56 Next, we want to evaluate the expectation of each of these spin matrices in this particular state.

01:05 So here are our definitions of our spin matrices, right, which terms of the poly matrices.

01:11 So when we compute sx, the average value of this, right, the average value of sx is going to be basically, let me write it this way, x, or kai, times xx times kai that is you know the definition of it for this particular state so when we do that our conjugate transpose is 2 negative 3 i we multiply this what this matrix does the s matrix basically flips the x and y components um and so what we're doing is computing it basically an inner product but where we flipped are the up and down components sorry not x and y the up and down components and so when we do that noting that this is the conjugate transpose that's just going to give us zero likewise the y matrix switches the up and down components but it negates the down components when it flips that and also multiplies them by factor of i and so when we do that it's effectively the same as flipping these components multiplying with you know the conjugate and so this whole thing comes out to zero once again.

02:28 You know, if you compute the center product, you can verify it for yourself.

02:32 And then the z matrix, this one just flips the down component, multiplying by h -bar over 2.

02:38 So when we do that, basically it's like, you know, taking each of these components squared and adding them together.

02:44 So we have like two squared plus negative three i squared, which is negative nine...

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