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| -rw-r--r-- | content/know/concept/electric-dipole-approximation/index.pdc | 147 | ||||
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diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc index bd8f76c..80707c6 100644 --- a/content/know/concept/einstein-coefficients/index.pdc +++ b/content/know/concept/einstein-coefficients/index.pdc @@ -136,115 +136,31 @@ This situation is mandatory for lasers, where stimulated emission must dominate, such that the light becomes stronger as it travels through the medium. -## Electric dipole approximation +## Coherent light -In fact, we can analytically calculate the Einstein coefficients, -if we make a mild approximation. -Consider the Hamiltonian of an electron with charge $q = - e$: - -$$\begin{aligned} - \hat{H} - &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V -\end{aligned}$$ - -With $\vec{A}(\vec{r}, t)$ the electromagnetic vector potential. -We reduce this by fixing the Coulomb gauge $\nabla \!\cdot\! \vec{A} = 0$, -such that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$, -and by assuming that $\vec{A}{}^2$ is negligible: - -$$\begin{aligned} - \hat{H} - &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{m} \vec{P} \cdot \vec{A} + V -\end{aligned}$$ - -The last term is the Coulomb interaction -between the electron and the nucleus. -We can interpret the second term, -involving the weak $\vec{A}$, as a perturbation $\hat{H}_1$: - -$$\begin{aligned} - \hat{H} - = \hat{H}_0 + \hat{H}_1 - \qquad \quad - \hat{H}_0 - \equiv \frac{\vec{P}{}^2}{2 m} + V - \qquad \quad - \hat{H}_1 - \equiv - \frac{q}{m} \vec{P} \cdot \vec{A} -\end{aligned}$$ - -Suppose that $\vec{A}$ is oscillating sinusoidally in time and space as follows: - -$$\begin{aligned} - \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) -\end{aligned}$$ - -The corresponding perturbative -[electric field](/know/concept/electric-field/) $\vec{E}$ -points in the same direction: - -$$\begin{aligned} - \vec{E}(\vec{r}, t) - = - \pdv{\vec{A}}{t} - = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) -\end{aligned}$$ - -Where $\vec{E}_0 = i \omega \vec{A}_0$. -Let us restrict ourselves to visible light, -whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$. -By comparison, the size of an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$, -so we can ignore the dot product $\vec{k} \cdot \vec{r}$. -This is the **electric dipole approximation**: -the radiation is treated classicaly, -while the electron is treated quantum-mechanically. - -$$\begin{aligned} - \vec{E}(\vec{r}, t) - \approx \vec{E}_0 \exp\!(- i \omega t) -\end{aligned}$$ - -Next, we want to convert $\hat{H}_1$ -to use the electric field $\vec{E}$ instead of the potential $\vec{A}$. -To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$ -and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/): - -$$\begin{aligned} - \matrixel{2}{\dv*{\vec{r}}{t}}{1} - &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1} - = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1} - \\ - &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1} - = i \omega_0 \matrixel{2}{\vec{r}}{1} -\end{aligned}$$ - -Therefore, $\vec{P} / m = i \omega_0 \vec{r}$, -where $\omega_0 = (E_2 - E_1) / \hbar$ is the resonance frequency of the transition, -close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating. -We thus get: +In fact, we can analytically calculate the Einstein coefficients in some cases, +by treating incoming light as a perturbation +to an electron in a two-level system, +and then finding $B_{12}$ and $B_{21}$ from the resulting transition rate. +We need to make the [electric dipole approximation](/know/concept/electric-dipole-approximation/), +in which case the perturbing Hamiltonian $\hat{H}_1(t)$ is given by: $$\begin{aligned} \hat{H}_1(t) - &= - \frac{q}{m} \vec{P} \cdot \vec{A} - = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) - \\ - &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) - = - \vec{p} \cdot \vec{E}_0 \exp\!(- i \omega t) + = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t) \end{aligned}$$ -Where $\vec{p} \equiv q \vec{r} = - e \vec{r}$ is the electric dipole moment of the electron, -hence the name *electric dipole approximation*. -Finally, because electric fields are actually real -(we made it complex for mathematical convenience), -we take the real part, yielding: +Where $q = -e$ is the electron charge, +$\vec{r}$ is the position operator, +and $\vec{E}_0$ is the amplitude of +the [electromagnetic wave](/know/concept/electromagnetic-wave-equation/). +For simplicity, we let the amplitude be along the $z$-axis: $$\begin{aligned} \hat{H}_1(t) - = - q \vec{r} \cdot \vec{E}_0 \cos\!(- i \omega t) + = - q E_0 z \cos\!(\omega t) \end{aligned}$$ - -## Polarized light - This form of $\hat{H}_1$ is a well-known case for [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), which tells us that the transition probability from $\ket{a}$ to $\ket{b}$ is: @@ -259,19 +175,20 @@ then generally $\ket{1}$ and $\ket{2}$ will be even or odd functions of $z$, such that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$, leading to: $$\begin{gathered} - \matrixel{1}{H_1}{2} = - q E_0 U + \matrixel{1}{H_1}{2} = - E_0 d \qquad - \matrixel{2}{H_1}{1} = - q E_0 U^* + \matrixel{2}{H_1}{1} = - E_0 d^* \\ \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0 \end{gathered}$$ -Where $U \equiv \matrixel{1}{z}{2}$ is a constant. +Where $d \equiv q \matrixel{1}{z}{2}$ is a constant, +namely the $z$-component of the **transition dipole moment**. The chance of an upward jump (i.e. absorption) is: $$\begin{aligned} P_{12} - = \frac{q^2 E_0^2 |U|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} + = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} \end{aligned}$$ Meanwhile, the transition probability for stimulated emission is as follows, @@ -280,7 +197,7 @@ and is therefore symmetric around $\omega_{ba}$: $$\begin{aligned} P_{21} - = \frac{q^2 E_0^2 |U|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} + = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} \end{aligned}$$ Surprisingly, the probabilities of absorption and stimulated emission are the same! @@ -289,8 +206,12 @@ the availability of electrons and holes in both states. In theory, we could calculate the transition rate $R_{12} = \pdv*{P_{12}}{t}$, which would give us Einstein's absorption coefficient $B_{12}$, -for this particular case of coherent monochromatic light. -However, the result would not be constant in time $t$. +for this specific case of coherent monochromatic light. +However, the result would not be constant in time $t$, +so is not really useful. + + +## Polarized light To solve this "problem", we generalize to (incoherent) polarized polychromatic light. To do so, we note that the energy density $u$ of an electric field $E_0$ is given by: @@ -301,11 +222,12 @@ $$\begin{aligned} E_0^2 = \frac{2 u}{\varepsilon_0} \end{aligned}$$ -Putting this in the previous result gives the following transition probability: +Where $\varepsilon_0$ is the vacuum permittivity. +Putting this in the previous result for $P_{12}$ gives us: $$\begin{aligned} P_{12} - = \frac{2 u q^2 |U|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} + = \frac{2 u |d|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} \end{aligned}$$ For a continuous light spectrum, @@ -313,11 +235,11 @@ this $u$ turns into the spectral energy density $u(\omega)$: $$\begin{aligned} P_{12} - = \frac{2 q^2 |U|^2}{\varepsilon_0 \hbar^2} + = \frac{2 |d|^2}{\varepsilon_0 \hbar^2} \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega} \end{aligned}$$ -From here, we the derivation is similar to that of +From here, the derivation is similar to that of [Fermi's golden rule](/know/concept/fermis-golden-rule/), despite the distinction that we are integrating over frequencies rather than states. @@ -329,8 +251,8 @@ which turns out to be $\pi t$: $$\begin{aligned} P_{12} - = \frac{q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x} - = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t + = \frac{|d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x} + = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t \end{aligned}$$ From this, the transition rate $R_{12} = B_{12} u(\omega_0)$ @@ -338,8 +260,8 @@ is then calculated as follows: $$\begin{aligned} R_{12} - = \pdv{P_{2 \to 1}}{t} - = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0) + = \pdv{P_{12}}{t} + = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \end{aligned}$$ Using the relations from earlier with $g_1 = g_2$, @@ -348,9 +270,9 @@ for a polarized incoming light spectrum: $$\begin{aligned} \boxed{ - B_{21} = B_{12} = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2} + B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} \qquad - A_{21} = \frac{\omega_0^3 q^2 |U|^2}{\pi \varepsilon \hbar c^3} + A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon \hbar c^3} } \end{aligned}$$ @@ -363,31 +285,33 @@ and define the polarization unit vector $\vec{n}$: $$\begin{aligned} \matrixel{1}{\hat{H}_1}{2} - = - q \matrixel{1}{\vec{r} \cdot \vec{E}_0}{2} - = - q E_0 \matrixel{1}{\vec{r} \cdot \vec{n}}{2} - = - q E_0 W + = - \vec{d} \cdot \vec{E}_0 + = - E_0 (\vec{d} \cdot \vec{n}) \end{aligned}$$ -The goal is to obtain the average of $|W|^2$, -where $W \equiv \matrixel{1}{\vec{r} \cdot \vec{n}}{2}$. +Where $\vec{d} \equiv q \matrixel{1}{\vec{r}}{2}$ is +the full **transition dipole moment** vector, which is usually complex. + +The goal is to calculate the average of $|\vec{d} \cdot \vec{n}|^2$. In [spherical coordinates](/know/concept/spherical-coordinates/), -we integrate over all possible orientations $\vec{n}$ for fixed $\vec{r}$, -using that $\vec{r} \cdot \vec{n} = |\vec{r}| \cos\!(\theta)$: +we integrate over all directions $\vec{n}$ for fixed $\vec{d}$, +using that $\vec{d} \cdot \vec{n} = |\vec{d}| \cos\!(\theta)$ +with $|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2$: $$\begin{aligned} - \expval{|W|^2} - = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\matrixel{1}{\vec{r}}{2}|^2 \cos^2(\theta) \sin\!(\theta) \dd{\varphi} \dd{\theta} + \expval{|\vec{d} \cdot \vec{n}|^2} + = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\vec{d}|^2 \cos^2(\theta) \sin\!(\theta) \dd{\varphi} \dd{\theta} \end{aligned}$$ Where we have divided by $4\pi$ (the surface area of a unit sphere) for normalization, -and $\theta$ is the polar angle between $\vec{n}$ and $\vec{p}$. +and $\theta$ is the polar angle between $\vec{n}$ and $\vec{d}$. Evaluating the integrals yields: $$\begin{aligned} - \expval{|W|^2} - = \frac{2 \pi}{4 \pi} |U|^2 \int_0^\pi \cos^2(\theta) \sin\!(\theta) \dd{\theta} - = \frac{|U|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi - = \frac{|U|^2}{3} + \expval{|\vec{d} \cdot \vec{n}|^2} + = \frac{2 \pi}{4 \pi} |\vec{d}|^2 \int_0^\pi \cos^2(\theta) \sin\!(\theta) \dd{\theta} + = \frac{|\vec{d}|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi + = \frac{|\vec{d}|^2}{3} \end{aligned}$$ With this additional constant factor $1/3$, @@ -395,16 +319,17 @@ the transition rate $R_{12}$ is modified to: $$\begin{aligned} R_{12} - = \frac{\pi q^2 |U|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0) + = \pdv{P_{12}}{t} + = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0) \end{aligned}$$ From which it follows that the Einstein coefficients for unpolarized light are given by: $$\begin{aligned} \boxed{ - B_{21} = B_{12} = \frac{\pi q^2 |U|^2}{3 \varepsilon_0 \hbar^2} + B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} \qquad - A_{21} = \frac{\omega_0^3 q^2 |U|^2}{3 \pi \varepsilon \hbar c^3} + A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon \hbar c^3} } \end{aligned}$$ diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc new file mode 100644 index 0000000..96b4fed --- /dev/null +++ b/content/know/concept/electric-dipole-approximation/index.pdc @@ -0,0 +1,147 @@ +--- +title: "Electric dipole approximation" +firstLetter: "E" +publishDate: 2021-09-14 +categories: +- Physics +- Quantum mechanics +- Optics + +date: 2021-09-14T13:11:54+02:00 +draft: false +markup: pandoc +--- + +# Electric dipole approximation + +Suppose that an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +is travelling through an atom, and affecting the electrons. +The general Hamiltonian of an electron in such a wave is given by: + +$$\begin{aligned} + \hat{H} + &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V +\end{aligned}$$ + +With charge $q = - e$ +and electromagnetic vector potential $\vec{A}(\vec{r}, t)$. +We reduce this by fixing the Coulomb gauge $\nabla \cdot \vec{A} = 0$, +so that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$, +and assume that $\vec{A}{}^2$ is negligible: + +$$\begin{aligned} + \hat{H} + = \hat{H}_0 + \hat{H}_1 + \qquad \quad + \hat{H}_0 + \equiv \frac{\vec{P}{}^2}{2 m} + V + \qquad \quad + \hat{H}_1 + \equiv - \frac{q}{m} \vec{P} \cdot \vec{A} +\end{aligned}$$ + +We have split $\hat{H}$ into $\hat{H}_0$ +and a perturbation $\hat{H}_1$, since $\vec{A}$ is small. +In an electromagnetic wave, +$\vec{A}$ is oscillating sinusoidally in time and space as follows: + +$$\begin{aligned} + \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) +\end{aligned}$$ + +The corresponding perturbative +[electric field](/know/concept/electric-field/) $\vec{E}$ +points in the same direction: + +$$\begin{aligned} + \vec{E}(\vec{r}, t) + = - \pdv{\vec{A}}{t} + = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) +\end{aligned}$$ + +Where $\vec{E}_0 = i \omega \vec{A}_0$. +Let us restrict ourselves to visible light, +whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$. +Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$, +so $\vec{k} \cdot \vec{r}$ is negligible: + +$$\begin{aligned} + \boxed{ + \vec{E}(\vec{r}, t) + \approx \vec{E}_0 \exp\!(- i \omega t) + } +\end{aligned}$$ + +This is the **electric dipole approximation**: +we ignore all spatial variation of $\vec{E}$, +and only consider its temporal oscillation. +Also, since we have not used the word "photon", +we are implicitly treating the radiation classically, +and the electron quantum-mechanically. + +Next, we want to convert $\hat{H}_1$ +to use the electric field $\vec{E}$ instead of the potential $\vec{A}$. +To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$ +and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/): + +$$\begin{aligned} + \matrixel{2}{\dv*{\vec{r}}{t}}{1} + &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1} + = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1} + \\ + &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1} + = i \omega_0 \matrixel{2}{\vec{r}}{1} +\end{aligned}$$ + +Therefore, $\vec{P} / m = i \omega_0 \vec{r}$, +where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition, +close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating. +We thus get: + +$$\begin{aligned} + \hat{H}_1(t) + &= - \frac{q}{m} \vec{P} \cdot \vec{A} + = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) + \\ + &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) + = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t) +\end{aligned}$$ + +Where $\vec{d} \equiv q \vec{r} = - e \vec{r}$ is +the **transition dipole moment operator** of the electron, +hence the name **electric dipole approximation**. +Finally, since electric fields are actually real +(we let it be complex for mathematical convenience), +we take the real part, yielding: + +$$\begin{aligned} + \boxed{ + \hat{H}_1(t) + = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t) + } +\end{aligned}$$ + +If this approximation is too rough, +$\vec{E}$ can always be Taylor-expanded in $(i \vec{k} \cdot \vec{r})$: + +$$\begin{aligned} + \vec{E}(\vec{r}, t) + = \vec{E}_0 \Big( 1 + (i \vec{k} \cdot \vec{r}) + \frac{1}{2} (i \vec{k} \cdot \vec{r})^2 + \: ... \Big) \exp\!(- i \omega t) +\end{aligned}$$ + +Taking the real part then yields the following series of higher-order correction terms: + +$$\begin{aligned} + \vec{E}(\vec{r}, t) + = \vec{E}_0 \Big( \cos\!(\omega t) + (\vec{k} \cdot \vec{r}) \sin\!(\omega t) - \frac{1}{2} (\vec{k} \cdot \vec{r})^2 \cos\!(\omega t) + \: ... \Big) +\end{aligned}$$ + + + +## References +1. M. Fox, + *Optical properties of solids*, 2nd edition, + Oxford. +2. D.J. Griffiths, D.F. Schroeter, + *Introduction to quantum mechanics*, 3rd edition, + Cambridge. diff --git a/content/know/concept/electromagnetic-wave-equation/index.pdc b/content/know/concept/electromagnetic-wave-equation/index.pdc index 68fe062..84946bb 100644 --- a/content/know/concept/electromagnetic-wave-equation/index.pdc +++ b/content/know/concept/electromagnetic-wave-equation/index.pdc @@ -118,14 +118,18 @@ $$\begin{aligned} \vb{E}(\vb{r}, t) &= \vb{E}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) \\ - \vb{H}(\vb{r}, t) - &= \vb{H}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \vb{B}(\vb{r}, t) + &= \vb{B}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) \end{aligned}$$ -In fact, thanks to linearity, these solutions can be treated as +In fact, thanks to linearity, these **plane waves** can be treated as terms in a Fourier series, meaning that virtually *any* function $f(\vb{k} \cdot \vb{r} - \omega t)$ is a valid solution. +Keep in mind that in reality, $\vb{E}$ and $\vb{B}$ are real, +so although it is mathematically convenient to use plane waves, +in the end you will need to take the real part. + ## Non-uniform medium diff --git a/content/know/concept/interaction-picture/index.pdc b/content/know/concept/interaction-picture/index.pdc new file mode 100644 index 0000000..1ce330d --- /dev/null +++ b/content/know/concept/interaction-picture/index.pdc @@ -0,0 +1,214 @@ +--- +title: "Interaction picture" +firstLetter: "I" +publishDate: 2021-09-13 +categories: +- Physics +- Quantum mechanics + +date: 2021-09-09T21:15:37+02:00 +draft: false +markup: pandoc +--- + +# Interaction picture + +The **interaction picture** or **Dirac picture** +is an alternative formulation of quantum mechanics, +equivalent to both the Schrödinger picture +and the [Heisenberg picture](/know/concept/heisenberg-picture/). + +Recall that Schrödinger lets states $\ket{\psi_S(t)}$ evolve in time, +but keeps operators $\hat{L}_S$ fixed (except for explicit time dependence). +Meanwhile, Heisenberg keeps states $\ket{\psi_H}$ fixed, +and puts all time dependence on the operators $\hat{L}_H(t)$. + +However, in the interaction picture, +both the states $\ket{\psi_I(t)}$ and the operators $\hat{L}_I(t)$ +evolve in $t$. +This might seem unnecessarily complicated, +but it turns out be convenient when considering +a time-dependent "perturbation" $\hat{H}_{1,S}$ +to a time-independent Hamiltonian $\hat{H}_{0,S}$: + +$$\begin{aligned} + \hat{H}_S(t) + = \hat{H}_{0,S} + \hat{H}_{1,S}(t) +\end{aligned}$$ + +With $\hat{H}_S(t)$ the full Schrödinger Hamiltonian. +We define the unitary conversion operator: + +$$\begin{aligned} + \hat{U}(t) + \equiv \exp\!\Big( i \frac{\hat{H}_{0,S} t}{\hbar} \Big) +\end{aligned}$$ + +The interaction-picture states $\ket{\psi_I(t)}$ and operators $\hat{L}_I(t)$ +are then defined to be: + +$$\begin{aligned} + \boxed{ + \ket{\psi_I(t)} + \equiv \hat{U}(t) \ket{\psi_S(t)} + \qquad + \hat{L}_I(t) + \equiv \hat{U}(t) \: \hat{L}_S(t) \: \hat{U}{}^\dagger(t) + } +\end{aligned}$$ + + +## Equations of motion + +To find the equation of motion for $\ket{\psi_I(t)}$, +we differentiate it and multiply by $i \hbar$: + +$$\begin{aligned} + i \hbar \dv{t} \ket{\psi_I} + &= i \hbar \Big( \dv{\hat{U}}{t} \ket{\psi_S} + \hat{U} \dv{t} \ket{\psi_S} \Big) + \\ + &= i \hbar \Big( i \frac{\hat{H}_{0,S}}{\hbar} \Big) \hat{U} \ket{\psi_S} + \hat{U} \Big( i \hbar \dv{t} \ket{\psi_S} \Big) +\end{aligned}$$ + +We insert the Schrödinger equation into the second term, +and use $\comm*{\hat{U}}{\hat{H}_{0,S}} = 0$: + +$$\begin{aligned} + i \hbar \dv{t} \ket{\psi_I} + &= - \hat{H}_{0,S} \hat{U} \ket{\psi_S} + \hat{U} \hat{H}_S \ket{\psi_S} + \\ + &= \hat{U} \big( \!-\! \hat{H}_{0,S} + \hat{H}_S \big) \ket{\psi_S} + \\ + &= \hat{U} \big( \hat{H}_{1,S} \big) \hat{U}{}^\dagger \hat{U} \ket{\psi_S} +\end{aligned}$$ + +Which leads to an analogue of the Schrödinger equation, +with $\hat{H}_{1,I} = \hat{U} \hat{H}_{1,S} \hat{U}{}^\dagger$: + +$$\begin{aligned} + \boxed{ + i \hbar \dv{t} \ket{\psi_I(t)} + = \hat{H}_{1,I}(t) \ket{\psi_I(t)} + } +\end{aligned}$$ + +Next, we do the same with an operator $\hat{L}_I$ +to find a description of its evolution in time: + +$$\begin{aligned} + \dv{t} \hat{L}_I + &= \dv{\hat{U}}{t} \hat{L}_S \hat{U}{}^\dagger + \hat{U} \hat{L}_S \dv{\hat{U}{}^\dagger}{t} + \hat{U} \dv{\hat{L}_S}{t} \hat{U}{}^\dagger + \\ + &= \frac{i}{\hbar} \hat{U} \hat{H}_{0,S} \big( \hat{U}{}^\dagger \hat{U} \big) \hat{L}_S \hat{U}{}^\dagger + - \frac{i}{\hbar} \hat{U} \hat{L}_S \big( \hat{U}{}^\dagger \hat{U} \big) \hat{H}_{0,S} \hat{U}{}^\dagger + + \Big( \dv{\hat{L}_S}{t} \Big)_I + \\ + &= \frac{i}{\hbar} \hat{H}_{0,I} \hat{L}_I + - \frac{i}{\hbar} \hat{L}_I \hat{H}_{0,I} + + \Big( \dv{\hat{L}_S}{t} \Big)_I + = \frac{i}{\hbar} \comm*{\hat{H}_{0,I}}{\hat{L}_I} + \Big( \dv{\hat{L}_S}{t} \Big)_I +\end{aligned}$$ + +The result is analogous to the equation of motion in the Heisenberg picture: + +$$\begin{aligned} + \boxed{ + \dv{t} \hat{L}_I(t) + = \frac{i}{\hbar} \comm*{\hat{H}_{0,I}(t)}{\hat{L}_I(t)} + \Big( \dv{t} \hat{L}_S(t) \Big)_I + } +\end{aligned}$$ + + +## Time evolution operator + +Recall that an alternative form of the Schrödinger equation is as follows, +where a **time evolution operator** or +**generator of translations in time** $K_S(t, t_0)$ +brings $\ket{\psi_S}$ from time $t_0$ to $t$: + +$$\begin{aligned} + \ket{\psi_S(t)} + = \hat{K}_S(t, t_0) \ket{\psi_S(t_0)} + \qquad \quad + \hat{K}_S(t, t_0) + \equiv \exp\!\Big( \!-\! i \frac{\hat{H}_S (t - t_0)}{\hbar} \Big) +\end{aligned}$$ + +We want to find an analogous operator in the interaction picture, satisfying: + +$$\begin{aligned} + \ket{\psi_I(t)} + \equiv \hat{K}_I(t, t_0) \ket{\psi_I(t_0)} +\end{aligned}$$ + +Inserting this definition into the equation of motion for $\ket{\psi_I}$ yields +an equation for $\hat{K}_I$, with the logical boundary condition $\hat{K}_I(t_0, t_0) = 1$: + +$$\begin{aligned} + i \hbar \dv{t} \Big( \hat{K}_I(t, t_0) \ket{\psi_I(t_0)} \Big) + &= \hat{H}_{1,I}(t) \Big( \hat{K}_I(t, t_0) \ket{\psi_I(t_0)} \Big) + \\ + i \hbar \dv{t} \hat{K}_I(t, t_0) + &= \hat{H}_{1,I}(t) \hat{K}_I(t, t_0) +\end{aligned}$$ + +We turn this into an integral equation +by integrating both sides from $t_0$ to $t$: + +$$\begin{aligned} + i \hbar \int_{t_0}^t \dv{t'} K_I(t', t_0) \dd{t'} + = \int_{t_0}^t \hat{H}_{1,I}(t') \hat{K}_I(t', t_0) \dd{t'} +\end{aligned}$$ + +After evaluating the left integral, +we see an expression for $\hat{K}_I$ as a function of $\hat{K}_I$ itself: + +$$\begin{aligned} + K_I(t, t_0) + = 1 + \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \hat{K}_I(t', t_0) \dd{t'} +\end{aligned}$$ + +By recursively inserting $\hat{K}_I$ once, we get a longer expression, +still with $\hat{K}_I$ on both sides: + +$$\begin{aligned} + K_I(t, t_0) + = 1 + \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} + + \frac{1}{(i \hbar)^2} \int_{t_0}^t \hat{H}_{1,I}(t') \int_{t_0}^{t'} \hat{H}_{1,I}(t'') \hat{K}_I(t'', t_0) \dd{t''} \dd{t'} +\end{aligned}$$ + +And so on. Note the ordering of the integrals and integrands: +upon closer inspection, we see that the $n$th term is +a [time-ordered product](/know/concept/time-ordered-product/) $\mathcal{T}$ +of $n$ factors $\hat{H}_{1,I}$: + +$$\begin{aligned} + \hat{K}_I(t, t_0) + &= 1 + \int_{t_0}^t \hat{H}_{1,I}(t_1) \dd{t_1} + + \frac{1}{2} \int_{t_0}^{t} \int_{t_0}^{t_1} \mathcal{T} \Big\{ \hat{H}_{1,I}(t_1) \hat{H}_{1,I}(t_2) \Big\} \dd{t_1} \dd{t_2} + + \: ... + \\ + &= 1 + \sum_{n = 1}^\infty \frac{1}{n!} \frac{1}{(i \hbar)^n} + \int_{t_0}^{t} \cdots \int_{t_0}^{t_n} \mathcal{T} \Big\{ \hat{H}_{1,I}(t_1) \cdots \hat{H}_{1,I}(t_n) \Big\} \dd{t_1} \cdots \dd{t_n} + \\ + &= \sum_{n = 0}^\infty \frac{1}{n!} \frac{1}{(i \hbar)^n} + \mathcal{T} \bigg\{ \bigg( \int_{t_0}^{t} \hat{H}_{1,I}(t') \dd{t'} \bigg)^n \bigg\} +\end{aligned}$$ + +Here, we recognize the Taylor expansion of $\exp$, +leading us to a final expression for $\hat{K}_I$: + +$$\begin{aligned} + \boxed{ + \hat{K}_I(t, t_0) + = \mathcal{T} \bigg\{ \exp\!\bigg( \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} \bigg) \bigg\} + } +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. + diff --git a/content/know/concept/lorentz-force/index.pdc b/content/know/concept/lorentz-force/index.pdc index f0e9850..2362766 100644 --- a/content/know/concept/lorentz-force/index.pdc +++ b/content/know/concept/lorentz-force/index.pdc @@ -231,7 +231,7 @@ Curiously, $\vb{v}_d$ is independent of $q$. Such a drift is not specific to an electric field. In the equations above, $\vb{E}$ can be replaced by a general force $\vb{F}/q$ (e.g. gravity) without issues. -In that case, $\vb{v}_d$ does depend on $q$. +In that case, $\vb{v}_d$ does depend on $q$: $$\begin{aligned} \boxed{ diff --git a/content/know/concept/time-ordered-product/index.pdc b/content/know/concept/time-ordered-product/index.pdc new file mode 100644 index 0000000..82c9d0f --- /dev/null +++ b/content/know/concept/time-ordered-product/index.pdc @@ -0,0 +1,124 @@ +--- +title: "Time-ordered product" +firstLetter: "T" +publishDate: 2021-09-13 +categories: +- Physics +- Quantum mechanics + +date: 2021-09-13T19:58:33+02:00 +draft: false +markup: pandoc +--- + +# Time-ordered product + +In quantum mechanics, especially quantum field theory, +a **time-ordered product** is a product of +explicitly time-dependent operators, +subject to certain ordering constraints. + +Let us start with an unusual motivation. +Suppose that some time-dependent operator $\hat{A}(t)$ is defined like so, +as a product of $N$ time-dependent sub-operators $\hat{a}_n(t)$: + +$$\begin{aligned} + \hat{A}(t) + \equiv \int_0^{t} \hat{a}_1(t_1) \bigg( \int_0^{t_1} \hat{a}_2(t_2) \bigg( \int_0^{t_2} \hat{a}_3(t_3) \bigg( \cdots \bigg) + \dd{t_3} \bigg) \dd{t_2} \bigg) \dd{t_1} +\end{aligned}$$ + +Crucially, the upper limits of the inner integrals +depend on the surrounding variables, +meaning that these integrals cannot simply be reordered. + +An interpretation is that the rightmost $\hat{a}_N(t_N)$ is applied first, +and then $\hat{a}_{N-1}(t_{N-1})$ secondly with $t_{N-1} > t_N$, +and so on. +This suggests there is a form of "time-ordering" here: +the integrals sweep across all relative timings of $\hat{a}_n$, +but preserve the ordering. +Indeed, this could be rewritten as a time-ordered product +(see the [interaction picture](/know/concept/interaction-picture/) for an example). + +A more general and intuitive motivation goes as follows. +Suppose we have a product of $N$ time-dependent operators $\hat{a}_n(t)$, +each representing a certain event. +Clearly, we would want to apply them in chronological order: + +$$\begin{aligned} + \hat{a}_N(t_N) \: \hat{a}_{N-1}(t_{N-1}) \: \cdots \: \hat{a}_2(t_2) \: \hat{a}_1(t_1) + \qquad \mathrm{where} \qquad + t_N > t_{N-1} > ... > \: t_2 > t_1 +\end{aligned}$$ + +But what if the ordering of the arguments $t_N, ..., t_1$ +is not known in advance? +We thus define the **time-ordering meta-operator** $\mathcal{T}$, +which reorders the operators based on the $t$-values +such that they are always in chronological order. +For example: + +$$\begin{aligned} + \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\} + \equiv + \begin{cases} + \hat{a}_1(t_1) \: \hat{a}_2(t_2) & \mathrm{if} \; t_2 < t_1 \\ + \hat{a}_2(t_2) \: \hat{a}_1(t_1) & \mathrm{if} \; t_1 < t_2 + \end{cases} +\end{aligned}$$ + +This example suggests a general algorithm for $\mathcal{T}$: +we need to consider every permutation of the operators $\hat{a}_n(t_n)$, +and leave only the single one that satisfies our demands. + +Mathematically, we do this by summing up all permutations, +and multiplying each term with a product of +[Heaviside step functions](/know/concept/heaviside-step-function/) $\Theta$, +which remove the term if the ordering is wrong: + +$$\begin{aligned} + \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\} + \equiv \sum_{p \in P_N}^{} + \Theta\big(t_{p_1} \!\!-\! t_{p_2}\big) \cdots \Theta\big(t_{p_{N-1}} \!\!-\! t_{p_N}\big) + \: \hat{a}_{p_1}(t_{p_1}) \: \cdots \: \hat{a}_{p_N}(t_{p_N}) +\end{aligned}$$ + +With this, our earlier example for two operators $\hat{a}_1$ and $\hat{a}_2$ +takes the following form: + +$$\begin{aligned} + \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\} + = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) + \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1) +\end{aligned}$$ + +However, we are still missing an important detail: +so far, we have quietly been assuming that the operators are bosonic +(see [second quantization](/know/concept/second-quantization/)). +To include fermionic operators, +we must allow the sign of each term to change, +based on whether the permutation is even or odd: + +$$\begin{aligned} + \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\} + = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) \pm \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1) +\end{aligned}$$ + +Where $\pm$ is $+$ for bosons, and $-$ for fermions in this case. +The general definition of $\mathcal{T}$ is: + +$$\begin{aligned} + \boxed{ + \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\} + \equiv \sum_{p \in P_N}^{} (-1)^p + \bigg( \prod_{j = 1}^{N-1} \Theta\big(t_{p_j} \!-\! t_{p_{j+1}}\big) \bigg) + \bigg( \prod_{k = 1}^N \hat{a}_{p_k}(t_{p_k}) \bigg) + } +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. |