Time dependence in quantum mechanics

Apr 28

Here I illustrate the difference between a ‘stationary’ and ‘non-stationary’ state in quantum mechanics and show why the former does not evolve in time.

Time evolution of the probability density for an initially localized particle.

The one-dimensional time-dependent Schrödinger equation (TDSE) is given by:

$i\hbar\frac{\partial\Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial\Psi(x,t)}{\partial x^2} + V(x) \Psi(x,t)$

For a time-independent Hamiltonian $\hat{H}$ the TDSE is separable, and the time-dependent wavefunction can therefore be written as a product of spatial and time-dependent factors,

$\Psi(x,t) = f(x) g(t),$

where $f(x)$ and $g (t)$ are functions of position and time, respectively. By substitution of the above expression into the TDSE we obtain,

$i\hbar f(x) \frac{\partial g(t)} {\partial t} = -\frac{\hbar^2}{2m} g(t) \frac{\partial f(x)}{\partial x^2} + V(x) f(x) g(t),$

and then divide both sides by $f(x)g(t)$ :

$\frac{i\hbar}{g(t)} \frac{\partial g(t)} {\partial t} = -\frac{\hbar^2}{2m f(x)}\frac{\partial f(x)}{\partial x^2} + V(x).$

The left hand side of the above equation depends only on time, and the right hand side depends only on position. Therefore, the above equation is true if and only if each side is equal to a constant $Z.$ The time-dependent part can be stated as,

$\frac{i\hbar}{g(t)} \frac{\partial g(t)} {\partial t} = Z$

and the position-dependent part as,

$-\frac{\hbar^2}{2m f(x)}\frac{\partial f(x)}{\partial x^2} + V(x) = Z.$

The solution to the time-dependent part is $g(t) = e^{\frac{-i\omega t}{\hbar},$ and the spatial part can be expressed in terms of the stationary states $\psi_m$ of the time-indepndent Hamiltonian $\hat{H}$ obtained by solving the time-independent Schrödinger equation (TISE). Expanded as a linear combination of stationary states of $\hat{H}$ , the time-dependent wavefunction is given by:

$\Psi(x,t) = \sum_{m}b_m\psi_me^{-iE_mt/\hbar}$

where $\psi_m$ are the stationary states obtained by solving the TISE, $b_m$ are the contributions of the stationary states to the superposition state at time $t=0$ , and the time evolution of the above equation is dictated by the eigenenergies $E_m$ .

Consider a single eigenstate $\psi_1$ of the TISE. As shown below, this state does not evolve in time:

$\psi_1\psi_1^*=\psi_1^{-iE_1t/\hbar}\psi_1^{+iE_1t/\hbar} = \psi_1^2$

and is therefore referred to as a ‘stationary state.’ Now suppose instead that the wavefunction is a superposition of two stationary states $\psi_1$ and $\psi_2$ ,

$\Psi(x,t) = b_1\psi_1^{-iE_1t/\hbar} + b_2\psi_2^{-iE_2t/\hbar}$

The probability density is given by:

$|\Psi(x,t)|^2 = (b_1\psi_1^{-iE_1t/\hbar} + b_2\psi_2^{-iE_2t/\hbar})(b_1\psi_1^{+iE_1t/\hbar} + b_2\psi_2^{+iE_2t/\hbar})$ $=b_1^2\psi_1^2+b_2^2\psi_2^2+b_1b_2\psi_1\psi_2e^{-i\omega t}+b_1b_2\psi_1\psi_2e^{i\omega t}$

where $\omega = E_2 - E_1$ and $\hbar = 1$ (in atomic units). By Euler’s identity $e^{i\theta}+e^{-i\theta} = 2\text{cos}(\theta)$ , the above expression is simplified to:

$|\Psi(x,t)|^2=b_1^2\psi_1^2+b_2^2\psi_2^2+2b_1b_2\psi_1\psi_2\text{cos}(\omega t)$

and the superposition state is shown to have time-dependence that varies sinusoidally as a function of the energy difference between the two contributing stationary states, $\psi_1$ and $\psi_2$ .