2: Foundations of Quantum Mechanics
- Page ID
- 467354
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- 2.1: The Schrödinger Equation
- Erwin Schrödinger posited an equation that predicts both the allowed energies of a system as well as address the wave-particle duality of matter. Schrödinger equation for de Broglie's matter waves cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof).
- 2.2: The Postulates of Quantum Mechanics
- There are only a small number of postulates of quantum mechanics. Upon them is built all of the conclusions of this powerful theory.
- 2.3: The Wavefunction Specifies the State of a System
- Postulate 1: Every physically-realizable state of the system is described in quantum mechanics by a state function that contains all accessible physical information about the system in that state.
- 2.4: Wavefunctions Have a Probabilistic Interpretation
- the most commonly accepted interpretation of the wavefunction that the square of the module is proportional to the probability density (probability per unit volume) that the electron is in the volume \(d\tau\) located at \(r_i\). Since the wavefunction represents the wave properties of matter, the probability amplitude \(P(x,t)\) will also exhibit wave-like behavior.
- 2.5: Wavefunctions Must Be Normalized
- To maintain the probabilistic interpretation of the wavefunction, the probability of a measurement of x yielding a result between -∞ and +∞ must be 1. Therefore, wavefunctions should be normalized (if possible) to ensure this requirement.
- 2.6: Linear Operators in Quantum Mechanics
- An operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another function.
- 2.7: Quantum Operators Represent Classical Variables
- Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. For an observable that is represented in classical physics by a function \(Q(x,p)\), the corresponding operator is \(Q(\hat{x},\hat{p})\).
- 2.8: The Schrödinger Equation is an Eigenvalue Problem
- To every dynamical variable in quantum mechanics, there corresponds an eigenvalue equation . The eigenvalues represents the possible measured values of the operator.
- 2.9: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators
- It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate.
- 2.10: Eigenfunctions of Operators are Orthogonal
- The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
- 2.11: The Heisenberg Uncertainty Principle
- The Heisenberg Uncertainty Principle is a fundamental theory in quantum mechanics that defines why a scientist cannot measure multiple quantum variables simultaneously. The principle asserts a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known.
- 2.12: Commuting Operators Allow Infinite Precision
- If two operators commute then both quantities can be measured at the same time with infinite precision, if not then there is a tradeoff in the accuracy in the measurement for one quantity vs. the other. This is the mathematical representation of the Heisenberg Uncertainty principle.