Fundamentals of Quantum Mechanics: Wave-like Behaviour, Uncertainty, and Classical Box Model

Wave-like Behaviour and Probability

Extended Nature of Waves:
- A wave, such as a water wave, occupies an extended space (e.g., an entire swimming pool).
- It's impossible to pinpoint its location; it exists simultaneously everywhere within its domain.
Quantum Particles as Waves:
- According to quantum mechanics, a quantum particle, if represented by a wave, also exists simultaneously everywhere.
Observation and Localization:
- When a quantum particle is observed, it is always found as a particle at a specific, well-defined location.
- Unlike classical particles, we cannot predict with certainty the future location of a quantum particle, even if its current location is known.
Probabilistic Nature:
- Repeated observations of the same quantum particle reveal that it is found more often in some areas and less often in others.
- This indicates that there is no certainty, only a probability, of finding the particle in a given area.
- For example, there might be a $90\%$ chance that an electron is within a certain distance from the nucleus.
Essential Probability Concepts (Appendix B):
- Probability distribution
- Average
- Distribution width
- These concepts underpin much of the course material and should be reviewed if unfamiliar.

Heisenberg Uncertainty Principle

Classical Prediction of Motion:
- Predicting the motion of everyday objects (e.g., tennis balls, rockets, planets) relies on simultaneously knowing both their momentum and location.
- Newton's equations use this information to accurately predict future positions (e.g., NASA's trajectory calculations).
Challenge for Quantum Particles:
- De Broglie's hypothesis states that a particle with a precise momentum has a precise wavelength ( $\lambda$ ).
- A quantum particle can be represented by an infinitely repeating wave (like a sine function), which extends to infinity, implying the particle could be located anywhere.
- This creates a conflict: if its momentum is precisely known (via wavelength), its position becomes infinitely uncertain.
Statement of Principle:
- The Heisenberg Uncertainty Principle states: "One cannot know precisely (with zero uncertainty) both the position and momentum of a microscopic particle at the same time."
Implications:
- This principle means that Newton's equations cannot be applied to microscopic particles that exhibit wave-like behavior.
- A new fundamental theory is required to predict the properties of particles with wave-like characteristics; this theory is quantum mechanics.
Notation:
- $\Delta x$ represents the uncertainty in position.
- $\Delta p$ represents the uncertainty in momentum.
Conceptual Connection: If wavelength $\lambda$ is precisely known, then momentum $p = \frac{h}{\lambda}$ is precisely known, but position ( $\Delta x$ ) is entirely uncertain. Conversely, if position is precisely known, then wavelength $\lambda$ and thus momentum $p$ ( $\Delta p$ ) become uncertain.

Classical Particle-in-a-box Model

Purpose: This model serves to highlight fundamental differences between classical and quantum mechanics.
Model Setup:
- A classical particle is confined to move in one dimension (e.g., along the x-axis) within a box.
- No external forces act on the particle while it is inside the box.
- The box is defined by infinitely high potential energy barriers (walls) at $x=0$ and $x=L$ , preventing the particle from escaping (Figure 4.6).
- Note (from handwritten annotations): This model avoids the complications of electromagnetism present in real atoms, which are 3D and involve electrostatic forces (e.g., between an electron and proton in hydrogen).
Classical Particle Behavior:
- The particle can be at rest anywhere within the box, or it can move back and forth, colliding elastically with the walls.
- If the system's energy is conserved, the particle moves with a constant speed between wall collisions.
Energy and Momentum:
- The total energy ( $E$ ) of the particle is solely kinetic energy:
  - $E = \frac{1}{2}mv^2$
  - This can be expressed in terms of momentum ( $p$ ) as Equation 4.4: $E = \frac{p^2}{2m}$
- Momentum is classically defined as $p = mv$ .
- If the particle is at rest, its velocity ( $v$ ), momentum ( $p$ ), and kinetic energy ( $E_k$ ) are all zero.
Distinction from Quantum: Crucially, an electron is NOT a classical particle; its wave-like properties must be considered.

Classical Probability Distribution in the Box

Classical Particle Properties (Summary):
- A classical particle can possess any continuous value for momentum and energy.
- It is equally likely to be found at any point within the box.
Probability Distribution:
- The probability distribution for a classical particle confined in a box is uniform across the entire length of the box.
- This probability is proportional to the inverse of the box's length ( $\frac{1}{L}$ ) (Figure 4.7).
- The particle's kinetic energy remains constant.

Exercises and Solutions

Exercise 1: Classical Particle in a One-Dimensional Box (x=0 to x=L) with Constant Kinetic Energy
- (a) Average position of the particle:
  - Given the uniform probability distribution, the particle is equally likely to be found at any point.
  - Therefore, the average position is at the center of the box: $\frac{L}{2}$ .
- (b) Average momentum of the particle:
  - The particle moves back and forth, spending equal time moving in the positive x-direction (positive momentum) and the negative x-direction (negative momentum).
  - Since momentum is a vector quantity, these equal and opposite momenta cancel out on average.
  - Thus, the average momentum is $0$ .
- (c) Total probability of finding the particle inside the box:
  - The particle is confined within the box and cannot escape.
  - Therefore, the total probability of finding the particle inside the box is $100\%$ . This corresponds to the total area under the probability distribution curve.