Phy5646/Grp3SpinProb

Worked Spin Problem

Consider a unit vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}} , measured an angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} from the positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{x}} axis in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} plane and angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} from the positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} axis.

Let the components of the spin vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{S}} along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}} be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = \hat{n} \cdot \overrightarrow{S}} .

a.) Solve the resulting eigenvalue equation. (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n \lambda = v \hbar \lambda } )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = S_x \sin(\theta) cos(\phi) + S_y \sin(\theta) \sin(\phi) + S_z \cos(\theta)}

Which, in matrix form (using the definitions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_x, S_y, S_z } ) looks like:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = \frac{\hbar}{2} \left(\begin{array}{cc} cos(\theta) & sin(\theta)e^{-i\phi} \\ sin(\theta)e^{i\phi} & -cos(\theta) \end{array} \right) }

And define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi = \left(\begin{array}{c} a \\ b \end{array} \right) } .

So that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n\chi = \frac{\hbar}{2} \left(\begin{array}{c} a \cos(\theta) + b \sin(\theta)e^{-i\phi} \\ a\sin(\theta)e^{i\phi} - b\cos(\theta) \end{array} \right) }

And the eigenvalue equation becomes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \begin{array}{c} a(\cos(\theta) - 2v) + b\sin(\theta)e^{-i\phi} \\ a\sin(\theta)e^{i\phi} - b(\cos(\theta) + 2v) \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right) }

Which has nontrivial solutions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \pm \frac{1}{2}}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_+, \frac{b}{a} = \tan(\frac{\theta}{2})e^{i\phi}}

Which means that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_+ = \left( \begin{array}{c} \cos(\frac{\theta}{2}) \\ \sin(\frac{\theta}{2})e^{-i\phi} \end{array} \right) }

And, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_-, \frac{b}{a} = -\cot(\frac{\theta}{2})e^{i\phi} }

So that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_- = \left( \begin{array}{c} \sin(\frac{\theta}{2}) \\ -\cos(\frac{\theta}{2})e^{i\phi} \end{array} \right) }

b.) Verify that the two resulting eigenvectors are orthogonal.

Show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\Chi_{+}|\Chi_{-}\rangle = 0 }

Where, from above,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\Chi_{+}| = \left( \cos(\frac{\theta}{2}) \;\; \sin(\frac{\theta}{2})e^{-i\phi} \right) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{-}\rangle = \left(\begin{array}{c} \sin(\frac{\theta}{2}) \\ -\cos(\frac{\theta}{2})e^{i\phi} \end{array} \right) }

So that,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\Chi_{+}|\Chi_{-}\rangle = \cos(\frac{\theta}{2})\sin(\frac{\theta}{2}) - \sin(\frac{\theta}{2})\cos(\frac{\theta}{2})e^{i\phi}e^{-i\phi} = 0}

As expected.

c.) Show that these vectors satisfy the closure relation.

This amounts to showing that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i} |\Chi_i\rangle\langle\Chi_i| = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) }

So:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\Chi_{+}| = \left(\cos(\frac{\theta}{2}) \; \sin(\frac{\theta}{2})e^{-i\phi}\right)}

And thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{+}\rangle\langle\Chi_{+}|= \left( \begin{array}{c} \cos(\frac{\theta}{2}) \\ \sin(\frac{\theta}{2})e^{-i\phi})\end{array} \right) \left(\cos(\frac{\theta}{2}) \sin(\frac{\theta}{2})e^{-i\phi}\right)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{+}\rangle\langle\Chi_{+}|= \left(\begin{array}{cc} \cos^2(\frac{\theta}{2}) & \sin(\frac{\theta}{2})\cos(\frac{\theta}{2})e^{-i\phi} \\ \sin(\frac{\theta}{2})\cos(\frac{\theta}{2})e^{-i\phi} & \sin^2(\frac{\theta}{2}) \end{array} \right) }

While

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{-}\rangle\langle\Chi_{-}|= \left( \begin{array}{c} \sin(\frac{\theta}{2}) \\ -\cos(\frac{\theta}{2})e^{i\phi})\end{array} \right) \left(\sin(\frac{\theta}{2}) -\cos(\frac{\theta}{2})e^{i\phi}\right)}

And Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{-}\rangle\langle\Chi_{-}|= \left(\begin{array}{cc} \sin^2(\frac{\theta}{2}) & -\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})e^{i\phi} \\ -\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})e^{i\phi} & \cos^2(\frac{\theta}{2}) \end{array} \right) }

So, clearly: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Chi_{-}\rangle\langle\Chi_{-}| + |\Chi_{+}\rangle\langle\Chi_{+}| = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) }