38-6 Schrödinger’s Equation

38-6 Schrödinger’s Equation#

Prompts

  • What is the wave function \(\Psi(x,y,z,t)\) for a matter wave? How is it separated into a space part \(\psi\) and a time part? What is the probability density and how do you compute it from \(\psi\)?

  • Write Schrödinger’s equation for a particle moving along the \(x\) axis with potential energy \(U(x)\). What is the angular wave number \(k\) in terms of \(E\), \(U\), \(m\), and \(h\)?

  • For uniform \(U\) (including \(U=0\)), Schrödinger’s equation becomes \(d^2\psi/dx^2 + k^2\psi = 0\). What is the general solution? What do the two terms represent (direction of motion)?

  • For a free particle (\(U=0\)), if we set \(B=0\) so \(\psi = Ae^{ikx}\), what is \(|\psi|^2\)? What does this imply about the particle’s location before measurement?

  • Given \(\psi\), how do you find the complex conjugate \(\psi^*\)? Why is \(|\psi|^2 = \psi\psi^*\) always real and nonnegative?

Lecture Notes#

Overview#

  • A matter wave is described by a wave function \(\Psi(x,y,z,t)\), which can be separated into a space-dependent part \(\psi\) and a time factor \(e^{-i\omega t}\).

  • Schrödinger’s equation (1926) determines \(\psi\) for a nonrelativistic particle. It is a basic principle—not derivable from more fundamental laws.

  • The probability density \(|\psi|^2\) gives the probability per unit volume of detecting the particle. The particle does not have a definite location until it is measured.

  • For uniform potential \(U\), the general solution is \(\psi = Ae^{ikx} + Be^{-ikx}\), with \(k\) related to momentum and kinetic energy.

The wave function#

Just as light waves are described by an electric field \(\vec{E}(x,y,z,t)\) and sound waves by pressure \(p(x,y,z,t)\), matter waves are described by a wave function \(\Psi(x,y,z,t)\) (uppercase psi). The wave function is generally complex—its values have the form \(a + ib\) with \(i^2 = -1\).

For the situations we consider, space and time separate:

(365)#\[ \Psi(x,y,z,t) = \psi(x,y,z)\, e^{-i\omega t} \]

where \(\omega = 2\pi f\) is the angular frequency. The spatial part \(\psi\) (lowercase psi) is what we find from Schrödinger’s equation.

Physical meaning: The probability of detecting the particle in a small volume centered at a point is proportional to \(|\psi|^2\) at that point. Because \(\psi\) is complex, we compute \(|\psi|^2 = \psi\psi^*\), where \(\psi^*\) is the complex conjugate of \(\psi\) (replace \(i\) by \(-i\) everywhere). The result is always real and nonnegative—it is the probability density.

Key point

A particle does not have a specific location until its position is actually measured. Before measurement, we only have probabilities given by \(|\psi|^2\).

Schrödinger’s equation#

Sound waves obey the wave equation from mechanics; light obeys Maxwell’s equations. Matter waves for nonrelativistic particles obey Schrödinger’s equation, proposed by Erwin Schrödinger in 1926. We cannot derive it from more basic principles—it is the basic principle.

For a particle of mass \(m\) moving along the \(x\) axis with total energy \(E\) and potential energy \(U(x)\):

(366)#\[ \frac{d^2\psi}{dx^2} + \frac{8\pi^2 m}{h^2}\bigl[E - U(x)\bigr]\psi = 0 \]

Note that \(E - U(x)\) is the kinetic energy at position \(x\).

For uniform potential energy \(U\) (constant, possibly zero), we define the angular wave number

(367)#\[ k = \frac{2\pi}{\lambda} = \frac{2\pi p}{h} = \frac{2\pi\sqrt{2m(E-U)}}{h} \]

where \(\lambda\) is the de Broglie wavelength and \(p\) is the momentum magnitude. Schrödinger’s equation simplifies to

(368)#\[ \frac{d^2\psi}{dx^2} + k^2\psi = 0 \]

General solution for uniform potential#

The general solution of Eq. (368) is

(369)#\[ \psi(x) = A e^{ikx} + B e^{-ikx} \]

where \(A\) and \(B\) are constants. Using Euler’s formula \(e^{i\theta} = \cos\theta + i\sin\theta\), the first term \(Ae^{ikx}\) represents a wave traveling in the \(+x\) direction; the second \(Be^{-ikx}\) represents a wave traveling in the \(-x\) direction.

To include time dependence, multiply by \(e^{-i\omega t}\):

(370)#\[ \Psi(x,t) = \psi(x)\, e^{-i\omega t} = A e^{i(kx-\omega t)} + B e^{-i(kx+\omega t)} \]

Probability density: free particle#

For a free particle (\(U=0\)) moving only in the \(+x\) direction, we set \(B=0\):

(371)#\[ \psi(x) = A e^{ikx} \]

The probability density is

(372)#\[ |\psi|^2 = |Ae^{ikx}|^2 = A^2 |e^{ikx}|^2 = A^2 \]

because \(|e^{ikx}|^2 = e^{ikx}(e^{ikx})^* = e^{ikx}e^{-ikx} = 1\). Thus \(|\psi|^2\) is constant—the particle has the same probability of being detected at any point along the \(x\) axis.

Interpretation

We cannot say the particle is “moving along” the axis like a classical object. Before measurement, it has no definite position; it could be found anywhere. The wave function describes a probability distribution, not a trajectory.

Summary#

  • Wave function \(\Psi = \psi\, e^{-i\omega t}\); spatial part \(\psi\) satisfies Schrödinger’s equation.

  • Schrödinger’s equation: \(d^2\psi/dx^2 + (8\pi^2 m/h^2)[E-U]\psi = 0\); for uniform \(U\), \(d^2\psi/dx^2 + k^2\psi = 0\).

  • Angular wave number \(k = 2\pi/\lambda = 2\pi\sqrt{2m(E-U)}/h\).

  • General solution (uniform \(U\)): \(\psi = Ae^{ikx} + Be^{-ikx}\).

  • Probability density \(|\psi|^2 = \psi\psi^*\); free particle has constant \(|\psi|^2\)—no definite location until measured.