11- CHEM-327 Lecture 11: Electron Wave Functions and Confinement

Electron States and Superposition of Momentum

Wave Function and Momentum Uncertainty

An electron, described by a wave function that is a superposition of various momentum values, is located at an unknown position and has an unknown momentum. This is a fundamental aspect of quantum mechanics where a precise position or momentum cannot be simultaneously known for such a state.

Superposition with a Range of Momentum Values

• Constructing a Wave Function: To represent an electron having a range of possible momenta, a superposition of momentum eigenstates is used.

• Mathematical Representation: This superposition is expressed as an integral, rather than a discrete sum, because we consider a continuous range of momentum values.

  • The general form of such a superposition is given by: \Psi(x) = N \int c(p) e^{i(px/\hbar)} dp where:

    • \Psi(x) is the wave function in position space.

    • N is a normalization constant, ensuring the total probability of finding the electron is one.

    • c(p) is the amplitude of the momentum eigenstate e^{i(px/\hbar)} corresponding to momentum p.

    • \hbar is the reduced Planck constant.

  • If c(p) is significant only for momenta between -p0 and p0, the integral would be over this finite range: \Psi(x) = N \int{-p0}^{p_0} e^{i(px/\hbar)} dp. This represents a state containing all momenta within that specified range.

• Determining Position: Evaluating this integral allows us to determine the position probability distribution of the electron. The resulting wave function, when plotted, typically shows a localized but not perfectly precise position.

Heisenberg's Uncertainty Principle

• Fundamental Principle: It can be rigorously shown that for any quantum mechanical system, an inherent limit exists on the precision with which certain pairs of physical properties, like position and momentum, can be simultaneously known.

• Mathematical Expression: This relationship is quantified by Heisenberg's Uncertainty Principle: \Delta x \Delta p \ge \frac{\hbar}{2} where:

  • \Delta x is the uncertainty in position.

  • \Delta p is the uncertainty in momentum.

  • \hbar (h/2\pi) is the reduced Planck constant.

• Implication: This principle signifies that a more precise measurement of position (smaller \Delta x) necessarily leads to a less precise knowledge of momentum (larger \Delta p), and vice-versa. This is not due to measurement inaccuracies but is an intrinsic property of quantum systems.

Uncertainty Principle Example

• Scenario: An electron is confined within a carbon nanotube of length L = 500 \text{ nm}.

• Problem: Calculate the minimum uncertainty in its momentum.

• Solution:

  • The uncertainty in position, \Delta x, is at least the length of the nanotube, 500 \text{ nm}.

  • Using Heisenberg's Uncertainty Principle:
    \Delta p \ge \frac{\hbar}{2\Delta x}

  • Substitute the values:

    • \hbar = 1.054 \times 10^{-34} \text{ J s}

    • \Delta x = 500 \text{ nm} = 500 \times 10^{-9} \text{ m} = 5 \times 10^{-7} \text{ m}
      \Delta p \ge \frac{1.054 \times 10^{-34} \text{ J s}}{2 \times (5 \times 10^{-7} \text{ m})}
      \Delta p \ge \frac{1.054 \times 10^{-34}}{10 \times 10^{-7}} \text{ N s}
      \Delta p \ge \frac{1.054 \times 10^{-34}}{10^{-6}} \text{ N s}
      \Delta p \ge 1.054 \times 10^{-28} \text{ N s}

  • Therefore, the minimum uncertainty in the electron's momentum is approximately 1.054 \times 10^{-28} \text{ N s}.

Particle in a Box Model

• Potential Energy Function: The behavior of a quantum mechanical system is heavily dependent on the potential energy function, V(x), used in the Schroedinger equation.

• Confinement: The