Quantum Mechanics#

  • Principles and postulates

    • statistics and measurements outcomes (Heisenberg built its matrix mechanics only on observables…)

    • CCR

  • angluar momentum, spin, and atom

Mathematical tools for quantum mechanics#

Definition 1 (Operator)

Definition 2 (Adjoint operator)

Given an operator \(\hat{A}: U \rightarrow V\), its adjoint operator \(\hat{A}^*: V \rightarrow U\) is the operator s.t.

\[(\mathbf{v}, \ \hat{A} \mathbf{u})_{V} = (\mathbf{u}, \hat{A}^* \mathbf{v} )_{U}\]

holds for \(\forall \mathbf{u} \in U, \ \mathbf{v} \in V\).

Definition 3 (Hermitian (self-adjoint) operator)

The operator \(\hat{A}: U \rightarrow U\) is a self-adjoint operator if

\[\hat{A}^* = \hat{A} \ .\]

Self-adjoint operators have real eigenvalues, and orthogonal eigenvectors (at least those associated to different eigenvalues; those associated with the same eigenvalues can be used to build an orthogonal set of vectors with orthogonalization process).

Postulates of Quantum Mechanics#

  • …

  • Canonical Commutation Relation (CCR) and Canonical Anti-Commutation Relation…

  • …

Non-relativistic Mechanics#

Statistical Interpretation and Measurement#

Wave function#

The state of a system is described by a wave function \(|\Psi\rangle\)

todo

  • properties: domain, image,…

  • unitary \(1 = \langle \Psi | \Psi \rangle = \left| \Psi \right|^2\), for statistical interpretation of \(\left| \Psi \right|^2\) as a density probability function

Operators and Observables#

Physical observable quantities are represented by Hermitian operators. Possible outcomes of measurement are the eigenvalues of the operator

Given \(\hat{A}\) and the set of its eigenvectors \(\{ |A_i \rangle \}_i\) (todo continuous or discrete spectrum…, need to treat this difference quite in details), with associated eigenvalues \(\{ a_i \}_i\)

\[\hat{A} |A_i \rangle = a_i |A_i\rangle\]
\[| \Psi \rangle = | A_i \rangle \langle A_i | \Psi \rangle = | A_i \rangle \Psi_i^A \]
\[\langle A_j | \Psi \rangle = \langle A_j | A_i \rangle \langle A_i | \Psi \rangle = \Psi_j^A \]

and thus

\[\begin{split}\begin{aligned} \Psi_j^A & = \langle A_j | \Psi \rangle \\ \Psi_j^{A*} & = \langle \Psi | A_j \rangle \\ \end{aligned}\end{split}\]

  • identity operator \(\sum_i | A_i \rangle \langle A_i | = \mathbb{I}\), since

    \[\sum_i | A_i \rangle \langle A_i | \Psi \rangle = \sum_i | A_i \rangle \langle A_i | \Psi_j^A A_j \rangle = \sum_i | A_i \rangle \delta_{ij} \Psi_j^A = \sum_i | A_i \rangle \Psi_i^A = | \Psi \rangle\]

  • Normalization:

    \[1 =\langle \Psi | \Psi \rangle = \Psi_j^{A*} \underbrace{\langle A_j | A_i \rangle}_{\delta_{ij}} \Psi_i^{A} = \sum_i \left| \Psi_i^A \right|^2\]

    with \(|\Psi_i^A|^2\) that can be interpreted as the probability of finding the system in state \(|\Psi_i^a\rangle\)

  • Expected value of the physical quantity in the a state \(|\Psi\rangle\), with possible values \(a_i\) with probability \(|\Psi_i^A|^2\)

    \[\begin{split}\begin{aligned} \bar{A}_{\Psi} & = \sum_i a_i |\Psi_i^A|^2 = \\ & = \sum_i a_i \Psi_i^{A*} \Psi_i^A = \\ & = \sum_i a_i \langle \Psi | A_i \rangle \langle A_i | \Psi \rangle = \\ & = \langle \Psi | \left( \sum_i a_i | A_i \rangle \langle A_i | \right) | \Psi \rangle = \\ & = \langle \Psi | \hat{A} | \Psi \rangle = \\ \end{aligned}\end{split}\]

    since an operator \(\hat{A}\) can be written as a function of its eigenvalues and eigenvectors

    \[\begin{split}\begin{aligned} \left( \sum_i a_i | A_i \rangle \langle A_i | \right) \Psi \rangle & = \left( \sum_i a_i | A_i \rangle \langle A_i | \right) c_k | A_k \rangle = \\ & = \sum_i a_i | A_i \rangle c_i = \\ & = \sum_i \hat{A} | A_i \rangle c_i = \\ & = \hat{A} \sum_i | A_i \rangle c_i = \hat{A} | \Psi \rangle \ . \end{aligned}\end{split}\]

Space Representation#

Position operator \(\hat{\mathbf{r}}\) has eigenvalues \(\mathbf{r}\) identifying the possible measurements of the position

\[\hat{\mathbf{r}} | \mathbf{r} \rangle = \mathbf{r} | \mathbf{r} \rangle \ ,\]

being \(\mathbf{r}\) the result of the measurement (position in space, mathematically it could be a vector), and \(| \mathbf{r} \rangle\) the state function corresponding to the measurement \(\mathbf{r}\) of the position.

  • Result of measurement, \(\mathbf{r}\), is a position in space. As an example, it could be a point in an Euclidean space \(P \in E^n\). It could be written using properties of Dirac’s delta “function”

    \[ \mathbf{r} = \int_{\mathbf{r}'} \delta (\mathbf{r}'-\mathbf{r}) \, \mathbf{r}' d \mathbf{r}' \]

  • Projection of wave function over eigenstates of position operator

    \[\begin{split}\begin{aligned} \langle \mathbf{r} | \Psi \rangle(t) = \Psi(\mathbf{r}, t) & = \int_{\mathbf{r}'} \delta(\mathbf{r} - \mathbf{r}') \Psi(\mathbf{r}',t) d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \langle \mathbf{r} | \mathbf{r}' \rangle \Psi(\mathbf{r}',t) d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \langle \mathbf{r} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi \rangle (t) d \mathbf{r}' = \\ & = \langle \mathbf{r} | \underbrace{\left( \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' | d \mathbf{r}' \right)}_{= \hat{\mathbf{1}}} | \Psi \rangle (t) \ . \end{aligned}\end{split}\]

  • having used orthogonality (todo why? provide definition and examples of operators with continuous spectrum)

    \[\langle \mathbf{r}' | \mathbf{r} \rangle = \delta(\mathbf{r}' - \mathbf{r})\]

  • Expansion of a state function \(|\Psi\rangle (t)\) over the basis of the position operator

    \[\begin{aligned} | \Psi \rangle (t) = \hat{\mathbf{1}} | \Psi \rangle(t) = \left( \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' d \mathbf{r}' \right) | \Psi \rangle(t) = \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi \rangle(t) \, d \mathbf{r}' \ . \end{aligned}\]

  • Unitariety and probability density

    \[\begin{split}\begin{aligned} 1 = \langle \Psi | \Psi \rangle (t) & = \langle \Psi | \left( \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' d \mathbf{r}' \right) | \Psi \rangle \\ & = \int_{\mathbf{r}'} \langle \Psi | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi \rangle \, d \mathbf{r}' \\ & = \int_{\mathbf{r}'} \Psi^*(\mathbf{r}',t) \Psi(\mathbf{r}',t) \, d \mathbf{r}' \\ & = \int_{\mathbf{r}'} \left| \Psi(\mathbf{r}',t) \right|^2 \, d \mathbf{r}' \\ \end{aligned}\end{split}\]

    and thus \(\left|\Psi(\mathbf{r},t)\right|^2\) can be interpreted as the probability density function of measuring position of the system equal to \(\mathbf{r}'\).

  • Average value of the operator

    \[\begin{split}\begin{aligned} \bar{\mathbf{r}} & = \langle \Psi | \hat{\mathbf{r}} | \Psi \rangle = \\ & = \int_{\mathbf{r}'} \langle \Psi | \mathbf{r}' \rangle \langle \mathbf{r}' | d \mathbf{r}' \ | \hat{\mathbf{r}} | \ \int_{\mathbf{r}''} | \mathbf{r}'' \rangle \langle \mathbf{r}'' | \Psi \rangle \, d \mathbf{r}'' \\ & = \int_{\mathbf{r}'} \int_{\mathbf{r}''} \langle \Psi | \mathbf{r}' \rangle \langle \mathbf{r}' | \hat{\mathbf{r}} | \mathbf{r}'' \rangle \langle \mathbf{r}'' | \Psi \rangle \, d \mathbf{r}' d \mathbf{r}'' = \\ & = \int_{\mathbf{r}'} \int_{\mathbf{r}''} \langle \Psi | \mathbf{r}' \rangle \underbrace{\langle \mathbf{r}' | \mathbf{r}'' \rangle}_{=\delta(\mathbf{r}'-\mathbf{r}'')} \mathbf{r}'' \langle \mathbf{r}'' | \Psi \rangle \, d \mathbf{r}' d \mathbf{r}'' = \\ & = \int_{\mathbf{r}'} \langle \Psi | \mathbf{r}' \rangle \mathbf{r}' \langle \mathbf{r}' | \Psi \rangle \, d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \Psi^*(\mathbf{r}',t) \ \mathbf{r}' \ \Psi(\mathbf{r}',t) \, d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \left| \Psi(\mathbf{r}',t) \right|^2 \, \mathbf{r}' \, d \mathbf{r}' \ . \end{aligned}\end{split}\]

Momentum Representation#

Momentum operator as the limit of…todo prove the expression of the momentum operator as the limit of the generator of translation

\[\langle \mathbf{r} | \hat{\mathbf{p}} = - i \hbar \nabla \langle \mathbf{r} | \]

  • Spectrum

    \[\hat{\mathbf{p}} | \mathbf{p} \rangle = \mathbf{p} | \mathbf{p} \rangle\]
    \[\langle \mathbf{r} | \hat{\mathbf{p}} | \mathbf{p} \rangle = - i \hbar \nabla \langle \mathbf{r} | \mathbf{p} \rangle = \mathbf{p} \langle \mathbf{r} | \mathbf{p} \rangle\]

    and thus the eigenvectors in space base \(\mathbf{p}(\mathbf{r}) = \langle \mathbf{r} | \mathbf{p} \rangle\) are the solution of the differential equation

    \[- i \hbar \nabla \mathbf{p}(\mathbf{r}) = \mathbf{p} \mathbf{p}(\mathbf{r}) \ ,\]

    that in Cartesian coordinates reads

    \[- i \hbar \partial_j p_k(\mathbf{r}) = p_j p_k(\mathbf{r})\]
    \[p_k(\mathbf{r}) = p_{k,0} \exp \left[ i \frac{p_j}{\hbar} r_j \right]\]

    or

    \[\langle \mathbf{r} | \mathbf{p} \rangle = \mathbf{p}(\mathbf{r}) = \mathbf{p}_0 \exp \left[ i \frac{\mathbf{p} \cdot \mathbf{r}}{\hbar} \right]\]

    todo

    • normalization factor \(\frac{1}{(2 \pi)^{\frac{3}{2}}}\)

      \[\mathscr{F}\{ \delta(x) \}(k) = \int_{-\infty}^{\infty} \delta(x) e^{-ikx} \, dx = 1\]

    • Fourier transform and inverse Fourier transform: definitions and proofs (link to a math section)

    • representation in basis of wave vector operator \(\hat{\mathbf{k}}\), \(\hat{\mathbf{p}} = \hbar \hat{\mathbf{k}}\)

From position to momentum representation#

Momentum and wave vector, \(\mathbf{p} = \hbar \mathbf{k}\)

\[\begin{split}\begin{aligned} \langle \mathbf{p} | \Psi \rangle & = \langle \mathbf{p} | \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \langle \mathbf{p} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ & = \frac{1}{(2\pi)^{3/2}} \int_{\mathbf{r}'} \exp \left[ i \frac{\mathbf{p} \cdot \mathbf{r}}{\hbar} \right] \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ \end{aligned}\end{split}\]

Relation between position and wave-number representation can be represented with a Fourier transform

\[\begin{split}\begin{aligned} \langle \mathbf{k} | \Psi \rangle & = \langle \mathbf{k} | \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ & = \int_{\mathbf{r}'} \langle \mathbf{k} | \mathbf{r}' \rangle \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ & = \frac{1}{(2\pi)^{3/2}} \int_{\mathbf{r}'} \exp \left[ i \mathbf{k} \cdot \mathbf{r}' \right] \langle \mathbf{r}' | \Psi\rangle d \mathbf{r}' = \\ & = \frac{1}{(2\pi)^{3/2}} \int_{\mathbf{r}'} \exp \left[ i \mathbf{k} \cdot \mathbf{r}' \right] \Psi(\mathbf{r}') d \mathbf{r}' = \\ & = \mathscr{F}\{ \Psi(\mathbf{r}) \} (\mathbf{k}) \end{aligned}\end{split}\]

Schrodinger Equation#

\[i \hbar \dfrac{d}{dt} | \Psi \rangle = \hat{H} | \Psi \rangle \]

being \(\hat{H}\) the Hamiltonian operator and \(|\Psi\rangle\) the wave function, as a function of time \(t\) as an independent variable.

Stationary States#

Eigenspace of the Hamiltonian operator

\[\hat{H} |\Psi_k \rangle = E_k |\Psi_k \rangle \ ,\]

with \(E_k\) possible values of energy measurements. If no eigenstates with the same eigenvalue exists, then…otherwise… Without external influence todo be more detailed!, energy values and eigenstates of the systems are constant in time.

Thus, exapnding the state of the system \(|\Psi\rangle\) over the stationary states gives \(|\Psi_k\rangle\), \(|\Psi\rangle = |\Psi_k\rangle c_k(t)\), and inserting in Schrodinger equation

\[i \hbar \dot{c}_k |\Psi_k\rangle = c_k E_k |\Psi_k\rangle\]

and exploiting orthogonality of eigenstates, a diagonal system for the amplitudes of stationary states ariese,

\[i \hbar \dot{c}_k = c_k E_k \ .\]

whose solution reads

\[c_k(t) = c_{k,0} \exp \left[- i \frac{E_k}{\hbar} t \right]\]

Thus the state of the system evolves like a superposition of monochromatic waves with frequencies \(\omega_k = \frac{E_k}{\hbar}\),

\[| \Psi \rangle = | \Psi_k \rangle c_k(t) = | \Psi_k \rangle c_{k,0}\exp\left[-i \frac{E_k}{\hbar} t \right] \ .\]
\[\begin{split}\begin{aligned} \dfrac{d}{dt} \bar{A} & = \dfrac{d}{dt} \left( \langle \Psi | \hat{A} | \Psi \rangle \right) = \\ & = \dfrac{d}{dt} \langle \Psi | \hat{A} | \Psi \rangle + \langle \Psi | \frac{d \hat{A}}{dt} | \Psi \rangle + \langle \Psi | \hat{A} \frac{d}{dt} | \Psi \rangle = \\ & = \langle \Psi | \frac{d \hat{A}}{dt} | \Psi \rangle + \frac{i}{\hbar} \langle \Psi | \hat{H} \hat{A} | \Psi \rangle - \frac{i}{\hbar} \langle \Psi | \hat{A} \hat{H} | \Psi \rangle = \\ & = \langle \Psi | \left( \frac{i}{\hbar} [ \hat{H}, \hat{A} ] + \frac{d \hat{A}}{dt} \right) | \Psi \rangle \ . \end{aligned}\end{split}\]

Pictures#

  • Schrodinger

  • Heisenberg

  • Interaction

Schrodinger#

If \(\hat{H}\) not function of time

\[| \Psi \rangle (t) = \exp\left[ - i \frac{\hat{H}}{\hbar} (t-t_0) \right] | \Psi \rangle(t_0) = \hat{U}(t,t_0) | \Psi \rangle(t_0) \]
\[\bar{A} = \langle \Psi | \hat{A} | \Psi \rangle = \langle \Psi_0 | \hat{U}^*(t,t_0) \hat{A} \hat{U}(t,t_0) | \Psi_0 \rangle\]
Heisenberg#

…

for \(\hat{H}\) independent from time \(t\),

\[\begin{split}\begin{aligned} \dfrac{d}{dt} \bar{\mathbf{r}} & = \overline{\frac{i}{\hbar} \left[ \hat{H}, \hat{\mathbf{r}} \right]} \\ \dfrac{d}{dt} \bar{\mathbf{p}} & = \overline{\frac{i}{\hbar} \left[ \hat{H}, \hat{\mathbf{p}} \right]} \\ \end{aligned}\end{split}\]
Hamiltonian Mechanics

From Lagrange equations

\[\dfrac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = Q_q\]

\(q\) generalized coordinates, \(p:= \frac{\partial L}{\partial \dot{q}}\) generalized momenta.

Hamiltonian

\[H(p,q,t) = p \dot{q} - L(\dot{q}, q, t)\]

Increment of the Hamiltonian

\[dH = \partial_p H dp + \partial_q H dq + \partial_t H dt \]
\[\begin{split}\begin{aligned} dH & = \dot{q} dp + p d \dot{q} - \partial_{\dot{q}} L d \dot{q} - \partial_{q} L d q - \partial_t L d t = \\ & = \dot{q} dp - \partial_{q} L d q - \partial_t L d t = \\ & = \dot{q} dp - \left( \dot{p} + Q_q \right) d q - \partial_t L d t = \\ \end{aligned}\end{split}\]
\[\begin{split}\begin{cases} \frac{\partial H}{\partial p} = \dot{q} \\ \frac{\partial H}{\partial q} = - \frac{\partial L}{\partial q} = - \dot{p} + Q_q \\ \frac{\partial H}{\partial t} = - \frac{\partial L}{\partial t} \end{cases}\end{split}\]

Physical quantity \(f(p(t), q(t), t)\). Its time derivative reads

\[\begin{split}\begin{aligned} \frac{d f}{dt} & = \frac{\partial f}{\partial p} \dot{p} + \frac{\partial f}{\partial q} \dot{q} + \frac{\partial f}{\partial t} = \\ & = \frac{\partial f}{\partial p} \left[ - \frac{\partial H}{\partial q} + Q_q \right] + \frac{\partial f}{\partial q} \frac{\partial H}{\partial p} + \frac{\partial f}{\partial t} = \\ & = \{ H, f \} + \partial_t f + Q_q \partial_p f \end{aligned}\end{split}\]

If \(Q_q = 0\), the correspondence between quantum mechanics and classical mechanics

\[\frac{d f}{d t} = \{ H, f \} + \partial_t f \qquad \leftrightarrow \qquad \dfrac{d}{dt} \overline{\hat{f}} = \overline{\frac{i}{\hbar} [ \hat{H}, \hat{f} ]} + \overline{\frac{\partial \hat{f}}{\partial t}}\]
\[\{ H, f \} \qquad \leftrightarrow \qquad \frac{i}{\hbar}[\hat{H}, \hat{f}]\]
Interaction#

Matrix Mechanics#

Attualization of 1925 papers#

…to find the canonical commutation relation,

\[[ \hat{\mathbf{r}}, \hat{\mathbf{p}} ] = i \hbar \mathbb{I} \, \hat{\mathbf{1}} \ .\]
\[\begin{split}\begin{aligned} \left[ \hat{\mathbf{r}}, \hat{\mathbf{p}} \right] & = \hat{\mathbf{r}} \hat{\mathbf{p}} - \hat{\mathbf{p}} \hat{\mathbf{r}} = \\ & = \hat{\mathbf{r}} \int_{\mathbf{r}} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r} \hat{\mathbf{p}} - \hat{\mathbf{p}} \int_{\mathbf{r}} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r} \ \hat{\mathbf{r}} \ \int_{\mathbf{r}'} | \mathbf{r}' \rangle \langle \mathbf{r}' | d \mathbf{r}' = \\ & = - \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle i \hbar \nabla \langle \mathbf{r} | \, d \mathbf{r} - \hat{\mathbf{p}} \int_{\mathbf{r}} \int_{\mathbf{r}'} | \mathbf{r} \rangle \mathbf{r}' \underbrace{\langle \mathbf{r} |\mathbf{r}' \rangle}_{\delta(\mathbf{r}-\mathbf{r}')} \langle \mathbf{r}' | d \mathbf{r}' = \\ & = - \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle i \hbar \nabla \langle \mathbf{r} | \, d \mathbf{r} - \hat{\mathbf{p}} \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r} = \\ & = - \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle i \hbar \nabla \langle \mathbf{r} | \, d \mathbf{r} - \int_{\mathbf{r}'} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r} \ \hat{\mathbf{p}} \ \int_{\mathbf{r}'} \mathbf{r}' | \mathbf{r}' \rangle \langle \mathbf{r}' | d \mathbf{r}' = \\ & = - \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle i \hbar \nabla \langle \mathbf{r} | \, d \mathbf{r} + \int_{\mathbf{r}} | \mathbf{r} \rangle i \hbar \nabla \langle \mathbf{r} | d \mathbf{r} \int_{\mathbf{r}'} \mathbf{r}' | \mathbf{r}' \rangle \langle \mathbf{r}' | d \mathbf{r}' = \dots \end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \left[ \hat{\mathbf{r}}, \hat{\mathbf{p}} \right] | \Psi \rangle & = - \int_{\mathbf{r}} \mathbf{r} | \mathbf{r} \rangle i \hbar \nabla \Psi(\mathbf{r},t) + \int_{\mathbf{r}} | \mathbf{r} \rangle i \hbar \nabla \left( \mathbf{r} \Psi(\mathbf{r},t) \right) = \\ & = - \int_{\mathbf{r}} | \mathbf{r} \rangle i \hbar \left[ \mathbf{r} \nabla \Psi(\mathbf{r},t) + \mathbb{I} \Psi(\mathbf{r},t) + \mathbf{r} \nabla \Psi(\mathbf{r},t) \right] = \\ & = i \hbar \underbrace{\int_{\mathbf{r}} | \mathbf{r} \rangle \langle \mathbf{r} | d \mathbf{r}}_{\hat{\mathbf{1}}} \, | \Psi \rangle \ , \end{aligned}\end{split}\]

and since \(| \Psi \rangle\) is arbitrary

\[\left[ \hat{\mathbf{r}}, \hat{\mathbf{p}} \right] = i \hbar \mathbb{I} \hat{\mathbf{1}} \ . \]
\[\left[ \hat{r}_a, \hat{p}_b \right] = i \hbar \delta_{ab} \ . \]

Heisenberg Uncertainty relation#

Uncertainty principle is a relation that holds for “wave descriptions” as it can be proved in the generic framework of Fourier transform, see Fourier transform:Uncertainty relation.

  • Heisenberg uncertainty relation is a relation between product of the variance of two physical quantities and their commutator,

  • todo relation with measurement process and outcomes. Commutation as a measurement process: first measure \(B\) and then \(A\), or first measure \(A\) and then \(B\)

\[\sigma_A \sigma_B \ge \frac{1}{2} \left|\overline{[\hat{A}, \hat{B}]}\right| \ .\]
Proof of Heisenberg uncertainty “principle”
\[\begin{split}\begin{aligned} \sigma^2_A \sigma^2_B & = \langle \Psi | \left(\hat{A} - \bar{A} \right)^2 | \Psi \rangle\langle \Psi | \left(\hat{B} - \bar{B} \right)^2 | \Psi \rangle = \\ & = \langle (\hat{A} - \bar{A} ) \Psi | (\hat{A} - \bar{A} ) \Psi \rangle \langle (\hat{B} - \bar{B} ) \Psi | (\hat{B} - \bar{B} ) \Psi \rangle = \\ & = \| ( \hat{A} - \bar{A} ) \Psi \|^2 \| ( \hat{B} - \bar{B} ) \Psi \|^2 = \\ & \ge \left| \langle ( \hat{A} - \bar{A} ) \Psi | ( \hat{B} - \bar{B} ) \Psi \rangle \right|^2 = \\ & = \left| \langle \Psi | (\hat{A} - \bar{A})(\hat{B} - \bar{B}) \Psi \rangle \right|^2 = \\ & = \left| \langle \Psi | \hat{A}\hat{B} - \hat{A}\bar{B} - \bar{A}\hat{B} + \bar{A}\bar{B} | \Psi \rangle \right|^2 = \\ & = \left| \langle \Psi | \hat{A}\hat{B} - \bar{A}\bar{B} | \Psi \rangle \right|^2 \ge && \text{(1)} \\ & = \left| \frac{\langle \Psi | \hat{A}\hat{B} - \hat{B}\hat{A} | \Psi \rangle}{2i} \right|^2 = \\ & = \frac{ \left|\langle \Psi | [\hat{A}, \hat{B}] | \Psi \rangle \right|^2}{4} = \frac{1}{4} \left| \overline{[\hat{A}, \hat{B}]} \right|^2 \end{aligned}\end{split}\]

having used Cauchy-Schwartz triangle inequality in (1),

\[|z| \ge | \text{im}(z) | = \frac{z - z^*}{2 i} \ .\]

Hesienberg uncertainty principles applied to position and momentum reads

\[\sigma_{r_a} \sigma_{p_b} \ge \frac{1}{2} \left|\overline{[\hat{r}_a, \hat{p}_b]}\right| = \frac{\hbar}{2} \delta_{ab} \ .\]

Many-body problem#

Wave function with symmetries: Fermions and Bosons