3: Postulates and Principles of Quantum Mechanics
- Page ID
- 424927
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- 3.1: 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.
- 3.2: 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.
- 3.3: 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.
- 3.4: 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 \(Q(\hat{x},\hat{p})\).
- 3.5: 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.
- 3.6: The Time-Dependent Schrödinger Equation
- While the time-dependent Schrödinger equation predicts that wavefunctions can form standing waves (i.e., stationary states), that if classified and understood, becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by the time-independent Schrödinger equation (used when the Hamiltonian is not explicitly time dependent). The solutions to the time-independent Schrödinger equation still have a time dependency.
- 3.E: Postulates and Principles of Quantum Mechanics (Exercises)
- These are homework exercises to accompany Chapter 4 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap.