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Diffstat (limited to 'source/know/concept/interaction-picture')
| -rw-r--r-- | source/know/concept/interaction-picture/index.md | 56 |
1 files changed, 28 insertions, 28 deletions
diff --git a/source/know/concept/interaction-picture/index.md b/source/know/concept/interaction-picture/index.md index 05f3ad0..de469fa 100644 --- a/source/know/concept/interaction-picture/index.md +++ b/source/know/concept/interaction-picture/index.md @@ -13,25 +13,25 @@ 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 $\hat{L}_H(t)$. +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 $t$. +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 $\hat{H}_{0,S}$: +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. +With $$\hat{H}_S(t)$$ the full Schrödinger Hamiltonian. We define the unitary conversion operator: $$\begin{aligned} @@ -41,7 +41,7 @@ $$\begin{aligned} } \end{aligned}$$ -The interaction-picture states $\Ket{\psi_I(t)}$ and operators $\hat{L}_I(t)$ +The interaction-picture states $$\Ket{\psi_I(t)}$$ and operators $$\hat{L}_I(t)$$ are then defined to be: $$\begin{aligned} @@ -57,8 +57,8 @@ $$\begin{aligned} ## Equations of motion -To find the equation of motion for $\Ket{\psi_I(t)}$, -we differentiate it and multiply by $i \hbar$: +To find the equation of motion for $$\Ket{\psi_I(t)}$$, +we differentiate it and multiply by $$i \hbar$$: $$\begin{aligned} i \hbar \dv{}{t}\Ket{\psi_I} @@ -68,7 +68,7 @@ $$\begin{aligned} \end{aligned}$$ We insert the Schrödinger equation into the second term, -and use $\comm{\hat{U}}{\hat{H}_{0,S}} = 0$: +and use $$\comm{\hat{U}}{\hat{H}_{0,S}} = 0$$: $$\begin{aligned} i \hbar \dv{}{t}\Ket{\psi_I} @@ -80,7 +80,7 @@ $$\begin{aligned} \end{aligned}$$ Which leads to an analogue of the Schrödinger equation, -with $\hat{H}_{1,I} = \hat{U} \hat{H}_{1,S} \hat{U}{}^\dagger$: +with $$\hat{H}_{1,I} = \hat{U} \hat{H}_{1,S} \hat{U}{}^\dagger$$: $$\begin{aligned} \boxed{ @@ -89,7 +89,7 @@ $$\begin{aligned} } \end{aligned}$$ -Next, we do the same with an operator $\hat{L}_I$ +Next, we do the same with an operator $$\hat{L}_I$$ to find a description of its evolution in time: $$\begin{aligned} @@ -120,8 +120,8 @@ $$\begin{aligned} 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 $t$: +**generator of translations in time** $$K_S(t, t_0)$$ +brings $$\Ket{\psi_S}$$ from time $$t_0$$ to $$t$$: $$\begin{aligned} \Ket{\psi_S(t)} @@ -138,8 +138,8 @@ $$\begin{aligned} \equiv \hat{K}_I(t, t_0) \Ket{\psi_I(t_0)} \end{aligned}$$ -Inserting this definition into the equation of motion for $\Ket{\psi_I}$ yields -an equation for $\hat{K}_I$, with the logical boundary condition $\hat{K}_I(t_0, t_0) = 1$: +Inserting this definition into the equation of motion for $$\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) @@ -150,7 +150,7 @@ $$\begin{aligned} \end{aligned}$$ We turn this into an integral equation -by integrating both sides from $t_0$ to $t$: +by integrating both sides from $$t_0$$ to $$t$$: $$\begin{aligned} i \hbar \int_{t_0}^t \dv{}{t'}K_I(t', t_0) \dd{t'} @@ -158,15 +158,15 @@ $$\begin{aligned} \end{aligned}$$ After evaluating the left integral, -we see an expression for $\hat{K}_I$ as a function of $\hat{K}_I$ itself: +we see an expression for $$\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}$$ -By recursively inserting $\hat{K}_I$ once, we get a longer expression, -still with $\hat{K}_I$ on both sides: +By recursively inserting $$\hat{K}_I$$ once, we get a longer expression, +still with $$\hat{K}_I$$ on both sides: $$\begin{aligned} K_I(t, t_0) @@ -175,9 +175,9 @@ $$\begin{aligned} \end{aligned}$$ 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 $\hat{H}_{1,I}$: +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 $$\hat{H}_{1,I}$$: $$\begin{aligned} \hat{K}_I(t, t_0) @@ -193,8 +193,8 @@ $$\begin{aligned} \end{aligned}$$ This construction is occasionally called the **Dyson series**. -We recognize the well-known Taylor expansion of $\exp(x)$, -leading us to a final expression for $\hat{K}_I$: +We recognize the well-known Taylor expansion of $$\exp(x)$$, +leading us to a final expression for $$\hat{K}_I$$: $$\begin{aligned} \boxed{ |