Time Evolution: From Classical to Quantum
Published:
Abstract: In this article I will introduce the quantum time evolution operator drawing parallels with classical mechanics and continuous transformations.
Prerequisites: Hamiltonian mechanics, Quantum mechanics (at an introductory level)
Classical mechanics and time evolution
In the Hamiltonian formalism of classical mechanics, we can consider groups of continuous transformations operating on a given system, starting from infinitesimal canonical transformations (canonical transformations which differ slightly from identity), in the general form:
\[Q_i = q_i + \epsilon \frac{\partial G}{\partial p_i} + o(\epsilon)\] \[P_i = p_i - \epsilon \frac{\partial G}{\partial q_i} + o(\epsilon)\]Generated by the functions $ F(q_i, p_i) = q_i p_i + \epsilon \ G(q_i, p_i)$ and $G = G(q_i, p_i)$
We can exchange $p_i$ and $P_i$ where the substitution is second order by working at first order in $\epsilon$
Preserved quantities, like angular momentum and energy, generate infinitesimal continuous transformations that correspond to continuous simmetries. For instance, angular momentum generates a continuous rotation around a given axis. Meanwhile, the Hamiltonian function $H(q_i, p_i)$ generates a particular transformation:
\[Q_i = q_i + \epsilon \frac{\partial H}{\partial p_i} + o(\epsilon)\] \[P_i = p_i - \epsilon \frac{\partial H}{\partial q_i} + o(\epsilon)\]And substituting with Hamilton’s equations gives:
\[Q_i = q_i + \epsilon \dot{q_i} + o(\epsilon)\] \[P_i = p_i + \epsilon \dot{p_i} + o(\epsilon)\]This is the time evolution of trajectories over phase space at first order! The Hamiltonian is then a function generating an ICT which maps points over phase space to the points of the same phase space after an “infinitesimal” time $\epsilon$ has passed, over the trajectories of motion:
\[q_i (t + \epsilon) = q_i(t) + \epsilon \dot{q_i}(t) + o(\epsilon)\] \[p_i (t + \epsilon) = p_i(t) + \epsilon \dot{p_i}(t) + o(\epsilon)\]This coherently comes around from the fact that the Hamiltonian function gives the time derivatives of the coordinates after derivation.
Classical time evolution operator
We can go even a step further and define an operator $\hat{H}=\{ \ \cdot \ , \ H \ \}$ which gives the time derivative of a function over phase space by the identity $\dot f = \{\ f \ , \ H \ \}$ (provided $f$ has no explicit time dependence). If we exponentiate this operator, we obtain a new operator which acts upon a function like this:
\[e^{t \hat H}f = (\ \sum_{n = 0}^{\infty} \frac{t^n}{n!} \hat{H}^n \ ) \ f \\ = f + t \ \{\ f \ , \ H \ \} + \frac{1}{2} t^2 \ \{ \ \{\ f \ , \ H \ \}, \ H \ \} \ + \ ... \\ = f + t \dot{f} + \frac{1}{2} t^2 \ddot{f} + \ ...\]In this power series, the functions should be evaluated at $t = 0$.
Applied to $q_i, p_i$, it gives back the previous result for small $t$ (the $\epsilon$ parameter is then time itself):
\[q_i(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!} (\frac{d^n}{dt^n} q_i)\,_{\rvert t = 0} \\ = q_i(0) + t \ \dot{q_i}(0) + o(t)\] \[p_i(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!} (\frac{d^n}{dt^n} p_i)\,_{\rvert t = 0} \\ = p_i(0) + t \ \dot{p_i}(0) + o(t)\]A similar idea is used to construct symplectic integrators in numerical methods.
The same operator can be applied to any function over phase space to obtain its time evolution, reminescent of quantum operator evolution.
Time evolution in quantum mechanics
In the formalism of quantum mechanics, on the other hand, we can consider the unitary operators generated by exponentiating self-adjoint operators, often called simmetry operators. Like in classical mechanics, the quantum angular momentum operator $\hat L_z$ generates a rotation around the $z$ axis:
\[R(\theta) = e^{-\frac{i}{\hbar} \theta \ \hat L_z}\]The Hamiltonian generates again a very interesting operator:
\[U(t) = e^{-\frac{i}{\hbar} t \hat H}\]This is exactly the time evolution operator and differs from its classical counterpart just by a scalar factor in the exponent which is typical of the change from classical to quantum mechanics (as immediately seen by the exchange of Poisson brackets and commutators).
Considering only their action on functions, the first “classical” operator $\hat H$ that we defined is analogous to the Hamiltonian operator of quantum mechanics, as functionally:
\[\hat H f = \frac{d}{dt} f\]which is almost equivalent to Schrodinger’s equation except for a factor $i \hbar$:
\[\hat H f = i \hbar \frac{\partial}{\partial t} f\]Schrodinger’s equation could be deduced by making suppositions on the time evolution of the system, while the $\hbar$ factor is introduced to adjust dimensionally.
This suggests a strong parallel between classical and quantum mechanics: the Hamiltonian is the generator of the evolution of the system in time.
Bibliography
- H. Goldstein, Classical Mechanics
- Sakurai & Napolitano, Modern Quantum Mechanics
- Charles Torre, Utah State University, Infinitesimal Canonical Transformations. Symmetries and Conservation Laws