Question

I am a novice to linear algebra fundamentals and quantum physics. From lecture today, my lecturer went over the importance of writing an operator in terms of the basis it reflecting in which it is quite trivial to setup

O^f = of
for a eigenstate.

We also have that we can define functions in a one-dimensional sense using projection
⟨x|g⟩=g(x)
such that typically one would perform this projection when designing the approach to their solution for their operator.

My difficulty lies in connecting these two concepts for operators like
O^=g(x^)×
such that imposing my first equation leads to a conundrum. Using my undergraduate study readings, I found a statement stating that operators of this form tend to share the same eigenvectors of |x⟩ or simply the basis of x. How does this make sense using the definition of eigenvalues/eigenvectors?

Answer 1

In quantum mechanics, operators act on wave functions to yield information about the system. Operators have associated eigenvalues and eigenvectors which represent measurable values and states of the system, respectively. The Hamiltonian operator is particularly important as it represents the total energy and is used in solving Schrödinger's equation.

Step-by-step explanation:

In quantum mechanics, the concept of operators is crucial for understanding the behavior of quantum systems. An operator is a mathematical object that, when applied to a wave function, yields some information about the system. For example, the position operator x gives us information about the particle's location. This is related to eigenvalues and eigenvectors because if a wave function is an eigenfunction of an operator, the corresponding eigenvalue is the value that is measured for the physical quantity associated with that operator.

The Hamiltonian operator Ĥ, which represents the total energy of a quantum particle, is crucial for solving the Schrödinger equation (Ĥy = Ey). The eigenvectors (wave functions) and eigenvalues (energy levels) obtained from this equation provide the special distribution of probabilities for finding the particle. The wave function must be normalized, and understanding how to work with operators includes applying them within the context of the wave function to extract useful physical information, like momentum or position expectation values, through a process of integration.

Understanding how operators act on wave functions and the significance of eigenvalues and eigenvectors is fundamental in quantum physics and is part of the broader topic of the quantum mechanical model. According to the model, quantum numbers are essential for characterizing states within a quantum system, and according to the Pauli exclusion principle, no two electrons can occupy the same quantum state. This underlines the uniqueness of quantum mechanical systems compared to classical systems.