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commutators_and_uncertainty_relations [2021/03/01 20:30] – created admincommutators_and_uncertainty_relations [2022/10/06 01:02] (current) – [Uncertainty Relations Between Operators] admin
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 The //**commutator**// of two operators $\hat{A}$ and $\hat{B}$ is The //**commutator**// of two operators $\hat{A}$ and $\hat{B}$ is
-\[\boxed{[\hat{A},\hat{B} = \hat{A}\hat{B}-\hat{B}\hat{A}.]}\]+\[\boxed{[\hat{A},\hat{B} ] = \hat{A}\hat{B}-\hat{B}\hat{A}.}\]
  
 Two operators //**commute**// if Two operators //**commute**// if
-\[[\hat{A}\hat{B}] = 0,\]+\[[\hat{A},\hat{B}] = 0,\]
 or, equivalently or, equivalently
 \[\hat{A}\hat{B} = \hat{B}\hat{A}.\] \[\hat{A}\hat{B} = \hat{B}\hat{A}.\]
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   * $[\hat{A}\hat{B},\hat{C}] = \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B}$   * $[\hat{A}\hat{B},\hat{C}] = \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B}$
   * The Jacobi identitiy: $\left [ \hat{A}, \left [ \hat{B},\hat{C} \right ]\right ] + \left [ \hat{B}, \left [ \hat{C},\hat{A} \right ]\right ] + \left [ \hat{C}, \left [ \hat{A},\hat{B} \right ]\right ] = 0$   * The Jacobi identitiy: $\left [ \hat{A}, \left [ \hat{B},\hat{C} \right ]\right ] + \left [ \hat{B}, \left [ \hat{C},\hat{A} \right ]\right ] + \left [ \hat{C}, \left [ \hat{A},\hat{B} \right ]\right ] = 0$
 +
 +Of these, the identities $[\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}] \hat{C} + \hat{B} [\hat{A},\hat{C}]$ and $[\hat{A}\hat{B},\hat{C}] = \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B}$ turn out to be very useful in calculations.  The way to remember them is that the operator that is on the left of the product comes out to the left and the operator that is on the right comes out to the right.  For example, in  $[\hat{A},\hat{B}\hat{C}]$, $\hat{B}$ is on the left of the product $\hat{B}\hat{C}$, so the term with $\hat{B}$ outside the commutator will be $\hat{B} [\hat{A},\hat{C}]$.  Similarly, $\hat{C}$ is on the right of the product $\hat{B}\hat{C}$, so it will come out on the right of the commutator in the term $[\hat{A},\hat{B}] \hat{C}$.  The same holds for the product $\hat{A}\hat{B}$ in the identity for $[\hat{A}\hat{B},\hat{C}]$.
  
 ====== Uncertainty Relations Between Operators ====== ====== Uncertainty Relations Between Operators ======
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 Since $\braket{\psi^{\perp}_B}{\psi^{\perp}_A}$ is the complex conjugate of $\braket{\psi^{\perp}_A}{\psi^{\perp}_B}$, we can rewrite this as Since $\braket{\psi^{\perp}_B}{\psi^{\perp}_A}$ is the complex conjugate of $\braket{\psi^{\perp}_A}{\psi^{\perp}_B}$, we can rewrite this as
 \[ \[
-  \Expect{[\hat{A},\hat{B}]} = 2 \Delta A \Delta B \mathrm{Im} \left ( \braket{\psi^{\perp}_A}{\psi^{\perp}_B} \right ). +  \Expect{[\hat{A},\hat{B}]} = 2i \Delta A \Delta B \mathrm{Im} \left ( \braket{\psi^{\perp}_A}{\psi^{\perp}_B} \right ). 
 \] \]
  
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   - Prove that commutators satisfy the Jacobi identity:   - Prove that commutators satisfy the Jacobi identity:
   \[\left [ \hat{A}, \left [ \hat{B},\hat{C} \right ]\right ] + \left [ \hat{B}, \left [ \hat{C},\hat{A} \right ]\right ] + \left [ \hat{C}, \left [ \hat{A},\hat{B} \right ]\right ] = 0.\]   \[\left [ \hat{A}, \left [ \hat{B},\hat{C} \right ]\right ] + \left [ \hat{B}, \left [ \hat{C},\hat{A} \right ]\right ] + \left [ \hat{C}, \left [ \hat{A},\hat{B} \right ]\right ] = 0.\]
-  - **The Aharonov-Vaidman identity:** If $\hat{\psi}$ is normalized and $\hat{A}$ is Hermitian, prove that+  - **The Aharonov-Vaidman identity:** If $\ket{\psi}$ is normalized and $\hat{A}$ is Hermitian, prove that
   \[\hat{A} \ket{\psi} = \Expect{A}\ket{\psi} + \Delta A \ket{\psi^{\perp}},\]   \[\hat{A} \ket{\psi} = \Expect{A}\ket{\psi} + \Delta A \ket{\psi^{\perp}},\]
   where $\ket{\psi^{\perp}}$ is normalized and orthogonal to $\ket{\psi}$.   where $\ket{\psi^{\perp}}$ is normalized and orthogonal to $\ket{\psi}$.