The Schrödinger Equation
Stationary States and the Schrödinger Equation
In quantum mechanics, the state of a particle is described by a wavefunction \( \psi(x,t), \) which contains all measurable information about the system. Its time evolution is governed by the time-dependent Schrödinger equation:
$$ i\hbar \frac{\partial}{\partial t} \, \psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \, \psi(x,t) + V(x) \, \psi(x,t) $$This is a partial differential equation in both space and time, and in general it is difficult to solve directly. A key simplification comes from looking for solutions using separation of variables:
$$ \psi(x,t) = \psi_E(x) \phi(t) $$ $$ i\hbar \frac{1}{\phi} \frac{d \phi}{dt} = -\frac{\hbar^2}{2m} \frac{1}{\psi_E} \frac{d^2 \psi_E}{dx^2} + V(x) $$Now, the left side is a function of \(t\) alone, and the right side is a function of \(x\) alone. This equality can hold for all \( x \) and \( t \) only if both sides are constant. Otherwise, by varying \(t\), you could change the left side without touching the right side, and the two would no longer be equal. For reasons that will become apparent soon, we shall call the constant E. The time equation is
$$ i\hbar \frac{1}{\phi} \frac{d \phi}{dt} = E $$ $$ \frac{d \phi}{dt} = - \frac{iE}{\hbar} \phi $$ $$ \Rightarrow \, \phi(t) = c e^{-i E t / \hbar} $$but we might as well absorb the constant \(c\) into \(\psi\), since the quantity of interest is the product \( \psi \phi \).
$$ \phi(t) = e^{-i E t / \hbar} $$The spatial equation satisfies the time-independent Schrödinger equation:
$$ -\frac{\hbar^2}{2m} \frac{d^2 \psi_E}{dx^2} + V(x) \psi_E = E \, \psi_E $$This is an eigenvalue equation \( \, \hat{E} \, \psi_E = E \, \psi_E \, \) where \( \hat{E} \) is the energy operator and \( E \) is the (energy) eigenvalue.
The energy eigenstates \( \psi_E(x) \) are called stationary states, and each one has a simple time dependence:
$$ \psi(x,t) = \psi_E(x) e^{-iEt/\hbar} $$All time dependence is therefore contained in a pure complex phase factor.
Stationary states play a central role because they have several important properties:
• Each stationary state has a definite energy \( E \)
• The probability density
\(
|\psi(x,t)|^2 = |\psi_E(x)|^2
\)
is independent of time (hence the name stationary states)
• They form an orthogonal basis of solutions, this allows arbitrary wavefunctions to be decomposed into simpler energy eigenstates.
The last point is crucial. It means that any physically allowed wavefunction can be written as a superposition of stationary states:
$$ \psi(x,t) = \sum_n c_n \psi_{E,n}(x)e^{-iE_n t/\hbar} $$where the coefficients \( c_n \) are determined by the initial condition \( \psi(x,0) \) by Fourier's Trick.
$$ \psi(x,0) = \sum_{n=1}^{\infty} c_n \psi_{E,n}(x) $$ $$ c_n = \int_{-\infty}^{\infty} \psi_{E,n}^*(x)\psi(x,0)\,dx $$This reduces the original problem of solving a complicated time-dependent partial differential equation to:
1. Solving the time-independent equation for the stationary states
2. Expanding the initial condition in terms of these states
3. Evolving each component in time using a simple exponential factor
In this way, stationary states provide a complete and systematic method for solving quantum systems.
Infinite Square Well Potential
The infinite square well potential is
$$ V(x) = \begin{cases} 0 & 0 \leq x \leq L \\ \infty & \text{elsewhere} \end{cases} $$The particle is therefore confined to the interval [\( 0,L \)], since the infinite potential outside the well is inaccessible.
The boundary conditions are
$$ \psi(0,t) = \psi(L,t) = 0 $$The wavefunction must vanish at the walls of the well.
For a free particle, the energy operator is
$$ \hat{E} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m} \, \partial_x^2 $$The energy eigenvalue equation is
$$ \hat{E} \, \psi_E = E \, \psi_E $$and so
$$ -\frac{\hbar^2}{2m} \, \psi_E''(x) = E \, \psi_E(x) $$Rearranging gives
$$ \psi_E''(x) + \frac{2mE}{\hbar^2}\psi_E(x) = 0 $$ $$ \psi_E''(x) + k^2\psi_E(x) = 0 \quad \text{where} \quad k^2 = \frac{2mE}{\hbar^2} $$ $$ \psi_E''(x) = - k^2 \psi_E(x) $$\( - k^2 \) is negative, therefore
$$ \psi_E(x) = c_1 e^{ikx} + c_2 e^{-ikx} $$and the energy is
$$ E = \frac{\hbar^2 k^2}{2m} $$The spatial part of the boundary conditions satisfy
$$ \psi_E(0) = \psi_E(L) = 0 $$ $$ \psi_E(0) = c_1 + c_2 = 0 \quad \Rightarrow \quad c_1 = -c_2 $$ $$ \psi_E(L) = c_1\left(e^{ikL} - e^{-ikL}\right) = 0 $$This holds \( \Leftrightarrow \, c_1 = 0 \), which gives the trivial solution, or
$$ e^{2ikL} = 1 $$ $$ \Rightarrow 2kL = 2\pi n, \quad n \in \mathbb{Z} $$ $$ k_n = \frac{n\pi}{L} $$The boundary conditions therefore quantize the allowed wavelengths and energies.
$$ \psi_E(x) = c_1\left(e^{ikx} - e^{-ikx}\right) $$ $$ = 2ic_1 \sin(kx) $$ $$ = A\sin\left(\frac{n\pi x}{L}\right) $$The case \( k_0 = 0 \) gives only the trivial solution, and negative values give nothing new since
$$ \sin(-x) = -\sin(x) $$ $$ \therefore \, n = 1,2,3,\dots $$The allowed energies are
$$ E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{n^2\pi^2\hbar^2}{2mL^2} $$We must normalize \( \psi_E \, \):
$$ \int_{0}^{L} |\psi_E(x)|^2\,dx = 1 $$ $$ |A|^2 \int_{0}^{L} \sin^2\left(\frac{n\pi x}{L}\right)\,dx = 1 $$ $$ |A|^2 \frac{L}{2} = 1 $$Therefore
$$ |A|^2 = \frac{2}{L} $$It is simplest to choose the positive real root. Therefore, inside the well,
$$ \psi_{E,n}(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \quad n = 1,2,3,\dots $$The stationary states of the infinite square well are
$$ \psi_n(x,t) = \psi_{E,n}(x) e^{-iE_n t/\hbar} $$where \( \displaystyle E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{n^2 \hbar^2 \pi^2}{2mL^2} \)
The most general solution is
$$ \psi(x,t) = \sum_{n=1}^{\infty} c_n \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) e^{-iE_n t/\hbar} $$where the coefficients are
$$ c_n = \sqrt{\frac{2}{L}} \int_{0}^{L} \sin\left(\frac{n\pi x}{L}\right)\psi(x,0)\,dx $$The orthogonality of the stationary states isolates each coefficient individually. Each stationary state evolves independently in time through its own phase factor.
The first four stationary states of the infinite square well are shown below:
See plot code