# 2.1: Time-Evolution with a Time-Independent Hamiltonian

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The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE):

$i \hbar \frac {\partial} {\partial t} \psi ( \overline {r} , t ) = \hat {H} ( \overline {r} , t ) \psi ( \overline {r} , t ) \label{1.1}$

$$\hat{H}$$ is the Hamiltonian operator which describes all interactions between particles and fields, and determines the state of the system in time and space. $$\hat{H}$$ is the sum of the kinetic and potential energy. For one particle under the influence of a potential

$\hat {H} = - \frac {\hbar^{2}} {2 m} \hat {\nabla}^{2} + \hat {V} ( \overline {r} , t ) \label{1.2}$

The state of the system is expressed through the wavefunction $$\psi ( \overline {r} , t )$$. The wavefunction is complex and cannot be observed itself, but through it we obtain the probability density

$P = | \psi ( \overline {r} , t ) |^{2},$

which characterizes the spatial probability distribution for the particles described by $$\hat{H}$$ at time $$t$$. Also, it is used to calculate the expectation value of an operator $$\hat{A}$$

\begin{align} \langle \hat {A} (t) \rangle &= \int \psi^{*} ( \overline {r} , t ) \hat {A} \psi ( \overline {r} , t ) d \overline {r} \\[4pt] &= \langle \psi (t) | \hat {A} | \psi (t) \rangle \label{1.3} \end{align}

Physical observables must be real, and therefore will correspond to the expectation values of Hermitian operators ($$\hat {A} = \hat {A}^{\dagger}$$).

Our first exposure to time-dependence in quantum mechanics is often for the specific case in which the Hamiltonian $$\hat{H}$$ is assumed to be independent of time: $$\hat {H} = \hat {H} ( \overline {r} )$$. We then assume a solution with a form in which the spatial and temporal variables in the wavefunction are separable:

$\psi ( \overline {r} , t ) = \varphi ( \overline {r} ) T (t) \label{1.4}$

$i \hbar \frac {1} {T (t)} \frac {\partial} {\partial t} T (t) = \frac {\hat {H} ( \overline {r} ) \varphi ( \overline {r} )} {\varphi ( \overline {r} )} \label{1.5}$

Here the left-hand side is a function only of time, and the right-hand side is a function of space only ($$\overline {r}$$, or rather position and momentum). Equation \ref{1.5} can only be satisfied if both sides are equal to the same constant, $$E$$. Taking the right-hand side we have

$\frac {\hat {H} ( \overline {r} ) \varphi ( \overline {r} )} {\varphi ( \overline {r} )} = E \quad \Rightarrow \quad \hat {H} ( \overline {r} ) \varphi ( \overline {r} ) = E \varphi ( \overline {r} ) \label{1.6}$

This is the Time-Independent Schrödinger Equation (TISE), an eigenvalue equation, for which $$\varphi ( \overline {r} )$$ are the eigenstates and $$E$$ are the eigenvalues. Here we note that

$\langle \hat {H} \rangle = \langle \psi | \hat {H} | \psi \rangle = E,$

so $$\hat{H}$$ is the operator corresponding to $$E$$ and drawing on classical mechanics we associate $$\hat{H}$$ with the expectation value of the energy of the system. Now taking the left-hand side of Equation \ref{1.5} and integrating:

\begin{align} i \hbar \frac {1} {T (t)} \frac {\partial T} {\partial t} &= E \\[4pt] \left( \frac {\partial} {\partial t} + \frac {i E} {\hbar} \right) T (t) &= 0 \label{1.7} \end{align}

which has solutions like this:

$T (t) = \exp ( - i E t / \hbar ) \label{1.8}$

So, in the case of a bound potential we will have a discrete set of eigenfunctions $$\varphi _ {n} ( \overline {r} )$$ with corresponding energy eigenvalues $$E_n$$ from the TISE, and there are a set of corresponding solutions to the TDSE.

$\psi _ {n} ( \overline {r} , t ) = \varphi _ {n} ( \overline {r} ) \underbrace{\exp \left( - i E _ {n} t / \hbar \right)}_{\text{phase factor}} \label{1.9}$

Phase Factor

For any complex number written in polar form (such as $$re^{iθ}$$), the phase factor is the complex exponential factor ($$e^{iθ}$$). The phase factor does not have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator. That is

$\langle \phi |A|\phi \rangle = \langle \phi |e^{-i\theta}Ae^{i\theta}|\phi \rangle$

Since the only time-dependence in $$\psi _ {n}$$ is a phase factor, the probability density for an eigenstate is independent of time:

$P = \left| \psi _ {n} (t) \right|^{2} = \text {constant}.$

Therefore, the eigenstates $$\varphi ( \overline {r} )$$ do not change with time and are called stationary states.

However, more generally, a system may exist as a linear combination of eigenstates:

\begin{align} \psi ( \overline {r} , t ) &= \sum _ {n} c _ {n} \psi _ {n} ( \overline {r} , t ) \\[4pt] &= \sum _ {n} c _ {n} e^{- i E _ {n} t h} \varphi _ {n} ( \overline {r} ) \label{1.10} \end{align}

where $$c_n$$ are complex amplitudes, with

$\sum _ {n} \left| c _ {n} \right|^{2} = 1. \nonumber$

For such a case, the probability density will oscillate with time. As an example, consider two eigenstates

\begin{align} \psi ( \overline {r} , t ) &= \psi _ {1} + \psi _ {2} \nonumber \\[4pt] &= c _ {1} \varphi _ {1} e^{- i E _ {1} t / h} + c _ {2} \varphi _ {2} e^{- i E _ {2} t / h} \label{1.11} \end{align}

For this state the probability density oscillates in time as

\begin{align} P (t) & = | \psi |^{2} \nonumber \\[4pt] &= \left| \psi _ {1} + \psi _ {2} \right|^{2} \nonumber \\[4pt] & = \left| c _ {1} \varphi _ {1} \right|^{2} + \left| c _ {2} \varphi _ {2} \right|^{2} + c _ {1}^{*} c _ {2} \varphi _ {1}^{*} \varphi _ {2} e^{- i \left( \alpha _ {2} - \omega _ {1} \right) t} + c _ {2}^{*} c _ {1} \varphi _ {2}^{*} \varphi _ {1} e^{+ i \left( a _ {2} - \omega _ {1} \right) t} \nonumber \\[4pt] & = \left| \psi _ {1} \right|^{2} + \left| \psi _ {2} \right|^{2} + 2 \left| \psi _ {1} \psi _ {2} \right| \cos \left( \omega _ {2} - \omega _ {1} \right) t \label{1.12} \end{align}

where $$\omega _ {n} = E _ {n} / \hbar$$. We refer to this state of the system that gives rise to this time-dependent oscillation in probability density as a coherent superposition state, or coherence. More generally, the oscillation term in Equation \ref{1.12} may also include a time-independent phase factor $$\phi$$ that arises from the complex expansion coefficients.

As an example, consider the superposition of the ground and first excited states of the quantum harmonic oscillator. The basis wavefunctions, $$\psi _ {0} (x)$$ and $$\psi _ {1} (x)$$, and their stationary probability densities $$P _ {i} = \left\langle \psi _ {i} (x) | \psi _ {i} (x) \right\rangle$$ are If we create a superposition of these states with Equation \ref{1.11}, the time-dependent probability density oscillates, with $$\langle x (t) \rangle$$ bearing similarity to the classical motion. (Here $$c_0 = 0.5$$ and $$c_1 = 0.87$$.) 