20.4.4. Modern reinterpretation of Heisenberg mechanics

20.4.4.1. Heisenberg Equations of Motion in the Heisenberg Picture

Time derivative of an operator \(\hat{A}_H\) in Heisenberg picture satisfies the relation (13.2),

\[\dfrac{d \hat{A}_H}{dt} = \frac{1}{i \hbar} \left[ \hat{A}_H , \hat{H}_H \right] + \left( \partial_t \hat{A} \right)_H \ .\]

Applying this relation to the position and momentum operators, \(\hat{\mathbf{x}}_H\) and \(\hat{\mathbf{p}}_H\), Heisenberg found the quantum mechanics counterpart of the equations of motion in classical mechanics,

(20.5)\[\begin{split}\left\{ \begin{aligned} \dot{\hat{\mathbf{x}}}_H & = \frac{1}{i \hbar} \left[ \hat{\mathbf{x}}_H , \hat{H}_H \right] = \frac{\hat{\mathbf{p}}_H}{m} \\ \dot{\hat{\mathbf{p}}}_H & = \frac{1}{i \hbar} \left[ \hat{\mathbf{p}}_H , \hat{H}_H \right] = - \left( \nabla_{\mathbf{r}} V\left( \hat{\mathbf{r}}\right) \right)_H \\ \end{aligned} \right.\end{split}\]

Details

If \(\hat{H} = \frac{|\hat{\mathbf{p}|}^2}{2 m} + V\left(\hat{\mathbf{r}}\right)\),

\(\left[ \hat{\mathbf{x}}_H, \hat{H}_H \right] = i \hbar \frac{\hat{\mathbf{p}}_H}{m}\)

\[\begin{split}\begin{aligned} \left[ \hat{\mathbf{x}}_H, \hat{H}_H \right] & = \mathscr{U}_{t,t_0}^{\dagger} \left( \left[ \hat{\mathbf{x}}, \frac{\left|\hat{\mathbf{p}}\right|^2}{2m} \right] + \left[ \hat{\mathbf{x}}, V\left(\hat{\mathbf{r}}\right) \right] \right) \mathscr{U}_{t,t_0} = \\ & = \mathscr{U}_{t,t_0}^{\dagger} \left( i \hbar \frac{\hat{\mathbf{p}}}{m} \right) \mathscr{U}_{t,t_0} = \\ & = i \hbar \frac{\hat{\mathbf{p}}_H}{m} \ . \end{aligned}\end{split}\]

since

1. \(\hat{\mathbf{x}}\) and \(V(\hat{\mathbf{r}})\) commute, as it can be proved using power expansion - if required - of \(V(\hat{\mathbf{r}})\);

2. the commutator of \(\hat{\mathbf{x}}\) and the kinetic contribution of the Hamiltonian operator reads (sum over \(b\) repeated index),

   \[\begin{split}\begin{aligned} \left[ \hat{x}_a, \hat{p}_b \hat{p}_b \right] & = \hat{x}_a \hat{p}_b \hat{p}_b - \hat{p}_b \hat{x}_a \hat{p}_b + \hat{p}_b \hat{x}_a \hat{p}_b - \hat{p}_b \hat{x}_b \hat{p}_a = \\ & = \left[ \hat{x}_a , \hat{p}_b \right] \hat{p}_b + \hat{p}_b \left[ \hat{x}_a , \hat{p}_b \right] = \\ & = 2 i \hbar \delta_{ab} \hat{p}_b = \\ & = 2 i \hbar \hat{p}_a \ . \end{aligned}\end{split}\]

\(\left[ \hat{\mathbf{p}}_H, \hat{H}_H \right] = - i \hbar \left( \nabla_{\mathbf{r}} V\left( \hat{\mathbf{r}}\right) \right)_H\)

\[\begin{split}\begin{aligned} \left[ \hat{\mathbf{p}}_H, \hat{H}_H \right] & = \mathscr{U}_{t,t_0}^{\dagger} \left( \left[ \hat{\mathbf{p}}, \frac{\left|\hat{\mathbf{p}}\right|^2}{2m} \right] + \left[ \hat{\mathbf{p}}, V\left(\hat{\mathbf{r}}\right) \right] \right) \mathscr{U}_{t,t_0} = \\ \end{aligned}\end{split}\]

since, the commutator of the momentum and potential energy operator reads \(\left[ \hat{\mathbf{p}}, V(\hat{\mathbf{r}}) \right] = \)

\[\begin{split}\begin{aligned} \langle \mathbf{r} | \left[ \hat{\mathbf{p}}, V\left(\hat{\mathbf{r}}\right) \right] | \Psi \rangle & = \langle \mathbf{r} | \hat{\mathbf{p}} V\left(\hat{\mathbf{r}}\right) | \Psi \rangle - \langle \mathbf{r} | V\left(\hat{\mathbf{r}}\right) \hat{\mathbf{p}} | \Psi \rangle = \\ & = - i \hbar \nabla_{\mathbf{r}} \langle \mathbf{r} | V\left(\hat{\mathbf{r}}\right) | \Psi \rangle - V\left(\mathbf{r}\right) \langle \mathbf{r} | \hat{\mathbf{p}} | \Psi \rangle = \\ & = - i \hbar \nabla_{\mathbf{r}} \left( V(\mathbf{r}) \Psi(\mathbf{r}, t) \right) + i \hbar V(\mathbf{r}) \nabla_{\mathbf{r}} \Psi(\mathbf{r}, t) = \\ & = - i \hbar \nabla_{\mathbf{r}} V(\mathbf{r}) \, \Psi(\mathbf{r},t) = \\ & = \langle \mathbf{r} | - i \hbar \nabla_{\mathbf{r}} V\left( \hat{\mathbf{r}} \right) | \Psi \rangle \ . \end{aligned}\end{split}\]

as \(\langle \mathbf{r} | \hat{\mathbf{p}} \rangle = - i \hbar \nabla_{\mathbf{r}} \langle \mathbf{r} |\), and \(V\left( \hat{\mathbf{r}} \right) | \mathbf{r} \rangle = V( \mathbf{r} ) | \mathbf{r} \rangle\), and

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

20.4.4.2. Matrix components in energy basis of the equations of motion

The matrix components in energy basis of the equations of motion (20.5) read

(20.6)\[\begin{split}\left\{ \begin{aligned} i \omega_{jk} X_{jk}^H & = \frac{P_{jk}^H}{m} \\ i \omega_{jk} P_{jk}^H & = - \left( \nabla V \right)^{H}_{jk} \\ \end{aligned} \right.\end{split}\]

with \(\omega_{jk} := \frac{E_j - E_k}{\hbar}\).

Details

If the Hamiltonian is not an explicit function of time, \(\mathscr{U}_{t,0} = \exp\left[ -i \frac{\hat{H}}{\hbar} t \right]\), and thus

\[\begin{split}\begin{aligned} \left( \hat{\mathbf{x}}_H \right) & = \left( | j \rangle \langle j | \hat{\mathbf{x}}_H | k \rangle \langle k | \right) = \\ & = \left( | j \rangle \langle j | \mathscr{U}_{t,0}^{\dagger} \hat{\mathbf{x}} \mathscr{U}_{t,0} | k \rangle \langle k | \right) = \\ & = | j \rangle \langle k | \left[ \exp\left(- i \frac{E_k-E_j}{\hbar} t \right) \langle j | \hat{\mathbf{x}} | k \rangle \right] \end{aligned}\end{split}\]

i.e.

\[X^{H}_{jk}(t) = X^{S}_{jk} \exp\left( i \frac{E_j - E_k}{\hbar} t \right) \ .\]

The time-derivative of the matrix elements of the position operator in Heisenber picture reads

\[\dot{X}^{H}_{jk} = i \omega_{jk} X^{S}_{jk} \exp\left( i \omega_{jk} t \right) = i \omega_{jk} X^H_{jk} \ ,\]

with \(\omega_{jk} := \frac{E_j - E_k}{\hbar}\).

20.4.4.3. Equations of motion as Hamilton’s equations

The equations of motion (20.5), or (20.6), can be recast in the form of Hamilton’s equations

\[\begin{split} \left\{ \begin{aligned} \dot{\hat{\mathbf{q}}}_H & = \dfrac{\partial \mathsf{H}}{\partial \hat{\mathbf{p}}_H} \\ \dot{\hat{\mathbf{p}}}_H & =-\dfrac{\partial \mathsf{H}}{\partial \hat{\mathbf{q}}_H} \\ \end{aligned} \right. \end{split}\]

and

\[\begin{split} \left\{ \begin{aligned} \dot{X}^{H}_{jk} & = \dfrac{\partial \mathsf{H}}{\partial P^{H}_{jk}} \\ \dot{P}^{H}_{jk} & =-\dfrac{\partial \mathsf{H}}{\partial Q^{H}_{jk}} \\ \end{aligned} \right. \end{split}\]

with \(\mathsf{H} = \text{tr} \left( \hat{H} \right) = \sum_a \langle a | \hat{H} | a \rangle \).

Details

20.4.4.4. CCR and quantization rule

20.4.4.4.1. Modern approach. Matrix form of the CCR

Given the CCR

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

its matrix components are

(20.7)\[\begin{aligned} i \hbar \delta_{ab} \delta_{jk} & = \sum_{\ell} \left\{ X^{a}_{j \ell} P^{b}_{\ell k} - P^{b}_{j \ell} X^{a}_{\ell k} \right\} \ . \end{aligned}\]

Details

\[\begin{split}\begin{aligned} i \hbar \delta_{ab} \delta_{jk} & = \langle j | i \hbar \delta_{ab} | k \rangle = \\ & = \langle j | \left[ \hat{x}_a , \hat{p}_b \right] k \rangle = \\ & = \langle j | \hat{x}_a \hat{p}_b - \hat{p}_b \hat{x}_a | k \rangle = \\ & = \sum_{\ell} \left\{ \langle j | \hat{x}_a | \ell \rangle \langle \ell | \hat{p}_b | k \rangle - \langle j | \hat{p}_b | \ell \rangle \langle \ell | \hat{x}_a | k \rangle \right\} = \\ & = \sum_{\ell} \left\{ X^{a}_{j \ell} P^{b}_{\ell k} - P^{b}_{j \ell} X^{a}_{\ell k} \right\} \ . \end{aligned}\end{split}\]

…

20.4.4.4.2. From Bohr-Sommerfeld quantization to CCR

In old quantum mechanics - i.e. quantum mechanics before a quantum mechanics theory - Bohr-Sommerfeld quantization rule was

\[J = n h = \oint p_n d q_n = \oint p_n \dot{q}_n \, dt \]

Classical trajectory. Using Fourier series of a periodic trajectory,

\[q_n(t) = \sum_{\alpha} q_\alpha(n) e^{i \alpha \omega_n t} \ ,\]

the time derivative of \(q_n(t)\) reads

\[\begin{aligned} \dot{q}_n(t) & = \sum_{\alpha} i \alpha \omega_n q_{\alpha}(n) e^{i \alpha \omega_n t} \ , \end{aligned}\]

while momentum can be written as

\[p_n(t) = \sum_{\beta} p_\beta(n) e^{i \beta \omega_n t} \ ,\]

Using these expression in the Bohr-Sommerfeld quantization rule,

\[\begin{split}\begin{aligned} J & = \dots = \\ & = i \, 2 \pi \sum_{\alpha} \alpha p^*_{\alpha}(n) q_{\alpha}(n) \ , \end{aligned}\end{split}\]

so that the derivative w.r.t. \(J\) reads

\[1 = i \, 2 \pi \sum_{\alpha} \alpha \dfrac{\partial}{\partial J} \left( p^*_{\alpha}(n) q_{\alpha}(n) \right) \ . \]

Quantum reinterpretation. Let a function \(\Phi(n,\alpha)\), then

\[\begin{split}\begin{aligned} \Phi(n, \alpha) - \Phi(n - \alpha, \alpha) & \sim \alpha \partial_n \Phi(n, \alpha) + o(\alpha) = \\ & = \alpha h \partial_J \Phi(n, \alpha) + o(\alpha) \ , \end{aligned}\end{split}\]

and the Born-Kramers correspondence principle follows,

\[\frac{\Phi(n, \alpha) - \Phi(n - \alpha, \alpha) }{ h } \sim \alpha \dfrac{\partial \Phi}{\partial J} (n, \alpha) \ .\]

Let \(p_\alpha(n) = p(n,\alpha) = p_{n+\alpha,n}\), then the reinterpretation of the classical quantization rule gives

\[\begin{split}\begin{aligned} 1 & = i \, \frac{2 \pi}{h} \sum_{\alpha} \left\{ p^*_{n+\alpha,n} q_{n+\alpha,n} - p^*_{n,n-\alpha} q_{n,n-\alpha} \right\} = \\ & = i \, \frac{2 \pi}{h} \sum_{\alpha} \left\{ p^*_{n+\alpha,n} q_{n+\alpha,n} - \sum_\alpha p^*_{n,n-\alpha} q_{n,n-\alpha} \right\} = \\ & = i \, \frac{2 \pi}{h} \sum_{\alpha} \left\{ p_{n,n+\alpha} q_{n+\alpha,n} - \sum_\alpha q_{n,n-\alpha} p_{n-\alpha,n} \right\} = \\ & = \frac{1}{i \hbar} \sum_{\alpha} \left\{ q_{n,\alpha} p_{\alpha,n} - p_{n,\alpha} q_{\alpha,n} \right\} \end{aligned}\end{split}\]

These relations are nothing but the diagonal components of the matrix form (20.7) of the CCR. The out-of-diagonal components are identically zero, as P.Jordan proved with the following trick

How P.Jordan proved that out-of-diagonal components of the CCR are identically zero

…

20.4.4.5. CCR in Energy Basis and Ladenburg Dispersion Formula

Introducing the expression of matrix components of the momentum \(\hat{\mathbf{p}}_H = m \dot{\hat{\mathbf{x}}}_H\), \(P^a_{jk} = i m \omega_{jk} X^{a}_{jk}\) into the matrix form (20.7) of the CCR relation

\[\begin{split}\begin{aligned} i \hbar \delta_{ab} \delta_{jk} & = \sum_{\ell} \left\{ X^{a}_{j \ell} P^{b}_{\ell k} - P^{b}_{j \ell} X^{a}_{\ell k} \right\} = \\ & = \sum_{\ell} \left\{ i m \omega_{\ell k} X^{a}_{j \ell} X^{b}_{\ell k} - i m \omega_{j \ell} X^{b}_{j \ell} X^{a}_{\ell k} \right\} \ . \end{aligned}\end{split}\]

The diagonal components \(j = k\) are

\[\begin{split}\begin{aligned} i \hbar \delta_{ab} & = \sum_{\ell} \left\{ i m \omega_{\ell k} X^{a}_{k \ell} X^{b}_{\ell k} - i m \omega_{k \ell} X^{b}_{k \ell} X^{a}_{\ell k} \right\} = \\ \end{aligned}\end{split}\]

If \(a = b\),

\[\begin{split}\begin{aligned} i \hbar & = \sum_{\ell} \left\{ i m \omega_{\ell k} X^{a}_{k \ell} X^{a}_{\ell k} - i m \omega_{k \ell} X^{a}_{k \ell} X^{a}_{\ell k} \right\} = \\ & = \sum_{\ell} \left\{ i m \omega_{\ell k} X^{a}_{k \ell} X^{a \, *}_{k \ell} - i m \omega_{k \ell} X^{a}_{k \ell} X^{a \, *}_{k \ell} \right\} = \\ & = i 2 m \, \sum_{\ell} \omega_{\ell k} \left| X^{a}_{k \ell} \right|^2 \ , \end{aligned}\end{split}\]

so that

\[ \frac{2 m}{\hbar} \, \sum_{\ell} \omega_{\ell k} \left| X^{a}_{k \ell} \right|^2 = 1 \ . \]

todo CHECK if there’s a factor \(3\), summing over all the components indexed by \(a\)

Connection to Ladenburg Dispersion Formula. In classical optics/early quantum theory, Ladenburg’s quantum dispersion formula for the oscillator strength \(f_{km}\) associated with a transition \(k \to m\) is defined as:

\[f_{km} = \frac{2m \omega_{km}}{\hbar} |x_{km}|^2\]

The diagonal CCR condition directly yields the Thomas-Reiche-Kuhn (TRK) sum rule:

\[\sum_m f_{km} = 1\]

This showed that Heisenberg’s matrix mechanics naturally incorporated the empirically validated dispersion theory of Ladenburg, Kramers, and Kronig.

20.4.4.6. Application: The Linear Harmonic Oscillator

Linear harmonic oscillator

For the harmonic oscillator Hamiltonian \(\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega_0^2 \hat{x}^2\):

Equations of Motion. $\(\dot{\hat{x}} = \frac{\hat{p}}{m}, \quad \dot{\hat{p}} = -m\omega_0^2 \hat{x}\)$

Taking the second time derivative:

\[\ddot{\hat{x}} + \omega_0^2 \hat{x} = 0\]

In matrix components:

\[(-\omega_{mn}^2 + \omega_0^2) x_{mn} = 0\]

Thus, non-zero matrix elements \(x_{mn}\) can only exist if \(\omega_{mn} = \pm \omega_0\), meaning transitions only occur between adjacent levels: \(m = n \pm 1\).

Matrix Elements & Energy Spectrum. Using the diagonal CCR \(\frac{2m}{\hbar} \sum_m \omega_{mn} |x_{nm}|^2 = 1\):

\[\frac{2m\omega_0}{\hbar} \left( |x_{n, n+1}|^2 - |x_{n, n-1}|^2 \right) = 1\]

With the boundary condition \(x_{0,-1} = 0\) for the ground state:

\[|x_{n, n+1}|^2 = \frac{\hbar}{2m\omega_0} (n+1)\]

The energy matrix elements give the discrete eigenvalues:

\[E_n = \hbar \omega_0 \left( n + \frac{1}{2} \right)\]

20.4.4.7. Transition Probabilities as Dipole Matrix Elements

When an atom interacts with an electromagnetic field, the interaction Hamiltonian is dominated by the electric dipole coupling:

\[\hat{V}(t) = - \hat{\mathbf{d}} \cdot \mathbf{E}(t) = - q \hat{\mathbf{x}} \cdot \mathbf{E}_0 \cos(\omega t)\]

The probability amplitude for a transition from state \(|n\rangle\) to state \(|m\rangle\) is dictated by the matrix element of the position/dipole operator:

\[x_{mn} = \langle m | \hat{x} | n \rangle\]

The transition rate \(W_{n \to m}\) (spontaneous emission / absorption probability) is proportional to the square of the dipole matrix element:

\[W_{n \to m} \propto |\mathbf{d}_{mn}|^2 = q^2 |\langle m | \hat{\mathbf{x}} | n \rangle|^2\]

20.4.4.8. Perturbation Theory for Transition Probabilities in Weak Electric Fields

Consider a time-dependent perturbation \(\hat{V}(t) = \hat{W} \cos(\omega t) = -q \hat{\mathbf{x}} \cdot \mathbf{E}_0 \cos(\omega t)\).

20.4.4.8.1. First-Order Amplitude

Writing state expansion \(|\psi(t)\rangle = \sum_n c_n(t) e^{-\frac{i}{\hbar}E_n t} |n\rangle\), the equations for \(c_m(t)\) are:

\[i\hbar \dot{c}_m(t) = \sum_n \langle m | \hat{V}(t) | n \rangle c_n(t) e^{i \omega_{mn} t}\]

Assuming the system starts in initial state \(c_k(0) = 1\) and \(c_m(0) = 0\) for \(m \neq k\):

\[c_m^{(1)}(t) = -\frac{i}{\hbar} \int_0^t V_{mk}(t') e^{i \omega_{mk} t'} dt'\]

Substituting \(V_{mk}(t') = -q E_0 x_{mk} \frac{e^{i\omega t'} + e^{-i\omega t'}}{2}\):

\[c_m^{(1)}(t) = \frac{q E_0 x_{mk}}{2\hbar} \left[ \frac{1 - e^{i(\omega_{mk} + \omega)t}}{\omega_{mk} + \omega} + \frac{1 - e^{i(\omega_{mk} - \omega)t}}{\omega_{mk} - \omega} \right]\]

20.4.4.8.2. Resonant Approximation (Rotating Wave Approximation)

Near resonance (\(\omega \approx \omega_{mk}\) with \(E_m > E_k\)):

The term with denominator \((\omega_{mk} - \omega)\) dominates:

\[|c_m^{(1)}(t)|^2 \approx \frac{q^2 E_0^2 |x_{mk}|^2}{4\hbar^2} \frac{\sin^2\left(\frac{(\omega_{mk} - \omega)t}{2}\right)}{\left(\frac{\omega_{mk} - \omega}{2}\right)^2}\]

20.4.4.8.3. Second-Order Expansion for \(|c_n(t)|^2\)

To compute probability to second order without assuming immediate resonance, we expand \(c_k(t)\) and \(c_m(t)\):

\[|c_k(t)|^2 = 1 + c_k^{(1)}(t) + c_k^{(1)*}(t) + |c_k^{(1)}(t)|^2 + \dots\]

Since \(V_{kk} = 0\) for parity-symmetric unperturbed states (\(x_{kk} = 0\)), the first-order correction to the initial state \(c_k^{(1)}(t) = 0\).

The conservation of total probability yields:

\[|c_k(t)|^2 = 1 - \sum_{m \neq k} |c_m^{(1)}(t)|^2\]

This confirms that weak external monochromatic fields cause probability transitions whose rates are directly dictated by matrix elements \(x_{mk}\) of Heisenberg’s matrix mechanics.