4: Postulates and Principles of Quantum Mechanics
- Page ID
- 11781
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: The Wavefunction Specifies the State of a System
- This page details the first postulate of quantum mechanics, asserting that a system's state is represented by its wavefunction, \(\psi\). Unlike classical mechanics, quantum states cannot precisely define position and momentum due to the uncertainty principle. Valid wavefunctions must be square-integrable, single-valued, continuous, and finite. The text clarifies that while some wavefunctions may meet certain criteria, others, like one that approaches infinity, are unacceptable.
- 4.2: Quantum Operators Represent Classical Variables
- This page describes the correspondence principle in quantum mechanics, stating that every classical observable has a corresponding quantum operator. It discusses observables like position and momentum, represented by operators acting on the wavefunction. Key operators include those for kinetic energy, potential energy, and the Hamiltonian, which combines both energy types. Additionally, a table outlines various quantum operators alongside their observables, emphasizing their mathematical forms.
- 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators
- This page explores eigenvalue equations in quantum mechanics, detailing the Hamiltonian operator, Schrödinger Equation, and how observables are derived as eigenvalues from eigenstates. It discusses expectation values for position, momentum, and kinetic energy in one-dimensional systems, noting that the average position is central, with zero momentum for the ground state.
- 4.4: The Time-Dependent Schrödinger Equation
- This page clarifies the differences between the time-dependent and time-independent Schrödinger equations, highlighting their effects on wavefunctions. It details how the time-dependent equation shows wavefunction evolution, while the time-independent equation indicates stationary states with constant energies.
- 4.5: Eigenfunctions of Operators are Orthogonal
- This page explains Hermitian operators in quantum mechanics, highlighting that they correspond to experimental observables with real eigenvalues and orthogonal eigenstates. It discusses the orthogonality of eigenfunctions, proving that wavefunctions from different eigenvalues are orthogonal, illustrated with particle-in-a-box examples. The text also addresses degenerate eigenstates and their potential non-orthogonality, which can be rectified through the Gram-Schmidt Orthogonalization process.
- 4.6: Commuting Operators Allow Infinite Precision
- This page explains the Heisenberg Uncertainty Principle and commutation relations in quantum mechanics, highlighting the significance of operators and their commutation. It details conditions for operator commutation and illustrates with examples, particularly regarding angular momentum. The text concludes that non-commuting operators impose limitations on the uncertainties of measurable physical quantities.
- 4.E: Postulates and Principles of Quantum Mechanics (Exercises)
- This page delves into quantum mechanics, covering key concepts such as wavefunctions, operators, and their commutation relations, particularly in the context of angular momentum. It emphasizes the uncertainty principle, the role of Hermitian operators, and the time-dependent Schrödinger equation. The text discusses the implications of non-commuting operators, the conditions for eigenvalues, and boundary conditions affecting wavefunctions.