3: Fundamentals of Quantum Mechanics
- Page ID
- 15742
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The previous chapter serves as a useful introduction to many of the basic concepts of quantum mechanics. In this chapter, we shall examine these concepts in a more systematic fashion. For the sake of simplicity, we shall concentrate on one-dimensional systems.
- 3.1: Schrödinger's Equation
- This page discusses a dynamical system of a non-relativistic particle of mass \(m\) moving along the \(x\)-axis under a potential \(V(x)\). It describes the particle's state with a complex wavefunction \(\psi(x,t)\) that evolves via Schrödinger’s equation, allowing for a probability density interpretation where \(|\psi(x,t)|^{2}\) determines the likelihood of the particle's position at \(x\).
- 3.2: Normalization of the Wavefunction
- This page discusses the concept of probability in quantum mechanics, emphasizing that values range from 0 to 1, indicating impossible to certain outcomes. It describes the normalization condition for wavefunctions, which requires the integral of their squared magnitude to equal one, ensuring probability consistency. The page also explains that for Gaussian wavefunctions, the normalization constant is derived through integration and remains valid over time according to Schrödinger's equation.
- 3.3: Expectation Values (Averages) and Variances
- This page explains the expectation value and variance of a particle's position in quantum mechanics, highlighting the use of the probability density \(|\psi(x,t)|^2\). It defines the expectation value \(\langle x \rangle\) as the average position based on multiple measurements and variance \(\sigma_x^2\) as the measurement spread.
- 3.4: Ehrenfest's Theorem
- This page explains how to calculate the expectation value of momentum in quantum mechanics through Ehrenfest's theorem, highlighting the parallels between quantum and classical mechanics. It shows that the expectation value is derived from the time derivative of position and relates to probability current. The behavior of quantum particles closely resembles that of classical particles when potential variations are slow, especially when wavefunction spatial extent is negligible.
- 3.5: Operators
- This page discusses operators as key components in quantum mechanics, highlighting their properties like linearity and non-commutativity. It explains the significance of Hermitian operators for real expectation values, introduces the Hamiltonian operator for energy, and connects classical dynamical variables to quantum operators. Additionally, it leads to expressions for energy and presents the Schrödinger equation, emphasizing the foundational role of operators in the quantum realm.
- 3.6: Momentum Representation
- This page covers key concepts in quantum mechanics, focusing on Fourier's theorem for one-dimensional wavefunctions and the relationship between position and momentum representations. It introduces the Dirac delta-function, explains transformations between wavefunctions \(\psi(x,t)\) and \(\phi(p,t)\), and discusses the calculation of expectation values.
- 3.7: Heisenberg's Uncertainty Principle
- This page explores the mathematical basis of Heisenberg's uncertainty principle, detailing variances of Hermitian operators and the implications of non-commutation. It also covers Fourier's theorem, connecting wave functions to energy representations and illustrating the minimum uncertainty principle with Gaussian wave functions. The discussion emphasizes how non-Gaussian wave-packets result in greater uncertainty, reinforcing the fundamental limits of measurement in quantum mechanics.
- 3.8: Eigenstates and Eigenvalues
- This page explains eigenstates and eigenvalues of Hermitian operators in quantum mechanics, where eigenstates remain proportional after the operator acts on them, and eigenvalues are real. It covers orthogonality of eigenstates corresponding to different eigenvalues and discusses degeneracy, allowing multiple eigenstates to share an eigenvalue. The page highlights that any wavefunction can be expressed as a linear combination of orthogonal eigenstates, using coefficients from inner products.
- 3.9: Measurement
- This page focuses on the behavior of Hermitian operators and wavefunctions in quantum mechanics, highlighting the importance of eigenstates for definite measurement results and the conditions for simultaneous measurements of operators. It also discusses the normalization and properties of wavefunctions, particularly in relation to position and momentum.
- 3.11: Stationary States
- This page covers stationary states in quantum mechanics, defined as energy eigenstates with a unique eigenvalue. The wavefunction comprises a spatial component and a time-exponential factor, leading to a constant probability density. It is governed by the time-independent Schrödinger equation, with general wavefunctions expressed as linear combinations of energy eigenstates.
- 3.12: Exercises
- This page covers fundamental concepts in quantum mechanics such as wavefunctions, expectation values, and operators. It explains measurement implications, particle behavior in potential structures, and introduces key principles like the uncertainty principle. Key discussions include orthogonality of wavefunctions, constancy of expectation values for specific operators, and time-evolution expressions.