Global AI and Data Science

Greetings to the IBM community!

In our previous exploration, we ventured into the fascinating world of elastic wave propagation and its numerical solutions using the Finite Difference Method (FDM). By understanding how stress and velocity fields evolve in a 1D elastic medium, we established a foundation for working with wave equations. While elastic waves offer a physical model for wave propagation, quantum systems take us deeper into the probabilistic nature of particles, where eigenvalues represent measurable quantities, much like stress and velocity in the elastic wave equation.

In wave mechanics [1], we discussed wave superposition and interference. These concepts naturally extend to quantum mechanics. Quantum states are often superpositions of eigenstates, and measurement collapses them into specific, observable outcomes.

Eigenvalues in Quantum Systems

In quantum mechanics, observables such as momentum, position, or energy are represented by Hermitian operators . These operators act on the state vectors in a Hilbert space . The eigenvalue problem for an operator is expressed as:

Where:

• A^ is a Hermitian operator representing an observable.
• $|\psi ⟩$ is an eigenstate (or eigenvector) corresponding to the observable.
• $\lambda$ is the eigenvalue, representing the value measured when the system is in the state $|\psi ⟩$.

For quantum systems, the Hermitian nature of ensures that the eigenvalues are real numbers, which aligns with the fact that observable quantities must yield real values when measured.

Example: Energy Eigenvalues in the Schrödinger Equation

Let's consider the time-independent Schrödinger equation:

where H^ is the Hamiltonian operator representing the total energy of the system, correspond to the possible energy levels of the system, and the eigenstates describe the quantum states in which the system will remain unchanged under the action of the Hamiltonian.

Solutions: The solutions are typically sinusoidal functions (sine and cosine waves) in free space, which can be visualized as standing waves in confined systems. These wavefunctions form a basis for describing quantum states.

When a quantum particle is confined within a certain region (like an infinite potential well), the wavefunction must satisfy specific boundary conditions. For instance, the wavefunction must be zero at the boundaries of the potential well (the walls of the box).

• Axes: The vertical axis represents the energy levels​, and the horizontal axis represents different eigenstates​.
• Energy Levels: Horizontal lines indicate energy levels corresponding to each eigenstate.
• Eigenstates: Blue curves represent eigenstates associated with each energy level.
• Hamiltonian Action: The label at the top right shows the action of the Hamiltonian , which governs the system.

Mathematical Representation: The allowed wavefunctions, which are solutions to the Schrödinger equation under these boundary conditions, can take the form:

where:

• n is a positive integer (quantum number) representing the mode of oscillation.
• L is the length of the potential well.

This function describes a standing wave with half-wavelengths fitting within the well.

Quantization

Discrete Energy Levels: The boundary conditions imposed lead to quantization, meaning only specific energy levels are allowed for the particle. Each eigenstate corresponds to an eigenvalue , which represents the energy associated with that state. The relationship between energy and quantum number for a one-dimensional infinite potential well is given by:

​where:

• •  $\hslash$ is the reduced Planck's constant,

• •  m is the mass of the particle.

Implications for Quantum Systems

1. Energy Eigenvalues: The eigenvalues ​ correspond to the discrete energy levels of the quantum system. Only these quantized levels are permissible, illustrating the restriction imposed by the boundary conditions.

2. Eigenstates: The eigenstates represent stable configurations of the quantum system where the particle exhibits a well-defined energy level without transitioning to another state. They depict the probability distribution of finding the particle in specific locations within the well.

3. Physical Interpretation: This mathematical framework allows us to understand physical systems where particles exist in quantized states, which has profound implications for the behavior of electrons in atoms, phonons in solids, and photons in cavities, among others.

Measurement and Probability in Quantum Mechanics

When we measure an observable associated with an operator A^, the system is described by a state vector $|\psi ⟩$, and the measurement yields one of the eigenvalues of with a probability P(λi​).

If is a linear combination (or superposition) of the eigenstates of the operator A^, we write:

Where:

•  ∣ϕ_i​⟩ are the eigenstates of A^ such that A^∣ϕ_i​⟩=λ_i​∣ϕ_i​⟩.
•  c_i​ are complex coefficients, representing the contribution of each eigenstate ∣ϕi​⟩ to the overall state $|\psi ⟩$.
• The probability of measuring the eigenvalue λi​ is given by:

The coefficients ​ are found through the projection of $|\psi ⟩$ onto the eigenstates ∣ϕi​⟩:

Thus, the probability of measuring ​ becomes:

Since the state vector $|\psi ⟩$ must be normalized, the sum of the probabilities of all possible measurement outcomes is 1:

This formalism ensures that a quantum measurement always yields a well-defined outcome with a probabilistic distribution, governed by the superposition of the eigenstates of the operator associated with the measured observable.

Collapse of the Wavefunction

Upon measurement of the observable A^, the system collapses to the eigenstate corresponding to the measured eigenvalue λ_i​. Mathematically, the post-measurement state is given by:

Thus, after measuring λi​, the system is found in the eigenstate ∣ϕi​⟩ with certainty, meaning that any subsequent measurement of  A^ will yield the same eigenvalue λi​.

By understanding these concepts in depth, we gain insight into the fundamental nature of quantum systems and how they are mathematically modeled through eigenvalues and eigenstates. This framework not only illuminates the discrete energy levels that characterize quantum particles but also highlights the probabilistic nature of measurement in quantum mechanics. As we explore the interplay between mathematical formalism and physical interpretation, it is nice to share this with all of you.

Always attentive to any suggestions or comments, and let's keep continue to learn about this rich tapestry of quantum behavior, where probability and determinism coexist, shaping our understanding of the microscopic world. Kind regards.