Completeness Relation Change of Basis

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  1. << Changing Spin Bases with a Completeness Relation | Completeness Relations | Matrix Elements and Completeness Relations >>

• change of basis spin half completeness relation dirac notation
• assignment Matrix Elements and Completeness Relations

  assignment Homework

  Matrix Elements and Completeness Relations

  Completeness Relations

  Quantum Fundamentals 2023 (3 years)

  Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

  What if I want to calculate the matrix elements using a different basis??

  The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: \(\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y\)

  In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

  One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

  where \(I\) is the identity operator: \(I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}\). This effectively rewrite the \(\left|{+}\right\rangle \) in the \(\left|{\pm}\right\rangle _y\) basis.

  Find the top row matrix elements of the operator \(\hat{S}_y\) in the \(S_z\) basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

• group Changing Spin Bases with a Completeness Relation

  group Small Group Activity

  10 min.

  Changing Spin Bases with a Completeness Relation

  Quantum Fundamentals 2023 (3 years)

  Completeness Relations Quantum States

  Completeness Relations

  Students work in small groups to use completeness relations to change the basis of quantum states.
• assignment Phase 2

  assignment Homework

  Phase 2

  quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2023 (3 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]

  1. For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
  2. Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
• group Outer Product of a Vector on Itself

  group Small Group Activity

  30 min.

  Outer Product of a Vector on Itself

  Quantum Fundamentals 2023 (2 years)

  Projection Operators Outer Products Matrices

  Completeness Relations

  Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• assignment Unknowns Spin-1/2 Brief

  assignment Homework

  Unknowns Spin-1/2 Brief

  Quantum Fundamentals 2023 (3 years) With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \).

  1. Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
  2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
  3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
  4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
• group $|\pm\rangle$ Forms an Orthonormal Basis

  group Small Group Activity

  30 min.

  \(|\pm\rangle\) Forms an Orthonormal Basis

  Quantum Fundamentals 2023 (3 years)

  Cartesian Basis $S_z$ basis completeness normalization orthogonality basis

  Completeness Relations

  Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

  accessibility_new Kinesthetic

  10 min.

  Using Arms to Represent Time Dependence in Spin 1/2 Systems

  Quantum Fundamentals 2023 (2 years)

  Arms Representation quantum states time dependence Spin 1/2

  Arms Sequence for Complex Numbers and Quantum States

  Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• accessibility_new Spin 1/2 with Arms

  accessibility_new Kinesthetic

  10 min.

  Spin 1/2 with Arms

  Quantum Fundamentals 2023 (2 years)

  Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

  Arms Sequence for Complex Numbers and Quantum States

  Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

  group Small Group Activity

  30 min.

  Finding if \(S_{x}, \; S_{y}, \; and \; S_{z}\) Commute

  Quantum Fundamentals 2023 (3 years)
• face Time Evolution Refresher (Mini-Lecture)

  face Lecture

  30 min.

  Time Evolution Refresher (Mini-Lecture)

  Central Forces 2023 (3 years)

  schrodinger equation time dependence stationary states

  Quantum Ring Sequence

  The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• Quantum Fundamentals 2023 (3 years)

  1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}

     Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]

  2. Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
  3. Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]