3: Model Systems in Quantum Mechanics
- Page ID
- 455289
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- 3.1: The Energy of a Particle in a Box is Quantized
- The particle in the box model system is the simplest non-trivial application of the Schrödinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics.
- 3.2: The Free Electron Model
- Consider a long molecule that is a conjugated polyene. Kuhn (Kuhn, 1949) has suggested a model for the electrons involved in this π-bond system in which an electron is said to have a finite potential energy when it is "on" the molecule and an infinite potential energy when it is "off" the molecule. The model (known as the free electron model) is very much analogous to the particle in a box problem.
- 3.3: The Average Momentum of a Particle in a Box is Zero
- From the mathematical expressions for the wavefunctions and energies for the particle-in-a-box, we can answer a number of interesting questions. Key to addressing these questions is the formulation and use of expectation values. This is demonstrated in the module and used in the context of evaluating average properties (energy, position, and momentum of the particle in a box).
- 3.4: A Particle in a Three-Dimensional Box
- The 1D particle in the box problem can be expanded to consider a particle within a 3D box for three lengths \(a\), \(b\), and \(c\). When there is NO FORCE (i.e., no potential) acting on the particles inside the box. Motion and hence quantization properties of each dimension is independent of the other dimensions. This Module introduces the concept of degeneracy where multiple wavefunctions (different quantum numbers) have the same energy.
- 3.5: Tunneling
- Tunneling is a quantum mechanical phenomenon when a particle is able to penetrate through a potential energy barrier that is higher in energy than the particle’s kinetic energy. This amazing property of microscopic particles play important roles in explaining several physical phenomena including radioactive decay. Additionally, the principle of tunneling leads to the development of Scanning Tunneling Microscope (STM).
- 3.6: A Harmonic Oscillator Obeys Hooke's Law
- The simple harmonic oscillator, a nonrelativistic particle in a quadratic potential, is an excellent model for a wide range of systems in nature. Indeed, it was for this system that quantum mechanics was first formulated: the blackbody radiation formula of Planck.
- 3.7: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule
- Viewing the multi-body system as a single particle allows the separation of the motion: vibration and rotation, of the particle from the displacement of the center of mass. This approach greatly simplifies many calculations and problems.
- 3.8: The Harmonic Oscillator Approximates Molecular Vibrations
- The quantum harmonic oscillator is the quantum analog of the classical harmonic oscillator and is one of the most important model systems in quantum mechanics. This is due in partially to the fact that an arbitrary potential curve V(x) can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it
- 3.9: The Harmonic Oscillator Energy Levels
- In this section we contrast the classical and quantum mechanical treatments of the harmonic oscillator, and we describe some of the properties that can be calculated using the quantum mechanical harmonic oscillator model.
- 3.10: The Harmonic Oscillator Wavefunctions involve Hermite Polynomials
- The quantum-mechanical description of vibrational motion using the harmonic oscillator model will produce vibrational quantum numbers, vibrational wavefunctions, quantized vibrational energies, and a zero-point energy.
- 3.11: Hermite Polynomials are either Even or Odd Functions
- Hermite polynomials were defined by Laplace (1810) though in scarcely recognizable form, and studied in detail by Chebyshev (1859). Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new. They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.
- 3.12: Spherical Polar Coordinates
- The description of a rotating molecule in Cartesian coordinates would be very cumbersome. The problem is actually much easier to solve in spherical polar coordinates.
- 3.13: Potential Energy and the Hamiltonian
- Since there is no energy barrier to rotation, there is no potential energy involved in the rotation of a molecule. All of the energy is kinetic energy.
- 3.14: Solution to the Schrödinger Equation
- The time-independent Schrödinger equation can be written as follows.
- 3.15: Spherical Harmonics
- The solutions to rigid rotor Hamiltonian are very important in a number of areas in chemistry and physics. The eigenfunctions are known as the spherical harmonics and they appear in every problem that has spherical symmetry.
- 3.16: Angular Momentum
- The Spherical Harmonics are involved in a number of problems where angular momentum is important (including the Rigid Rotor problem, the H-atom problem and anything else where spherical symmetry is involved.)
- 3.17: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision
- The angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. Two orthogonal components of angular momentum (e.g., \(L_x\) and \(L_y\)) are complementary and cannot be simultaneously known or measured. It is, however, possible to simultaneously measure or specify \(L^2\) and any one component of \(L\).
- 3.18: Application to the Rotation of Real Molecules
- While the spherical harmonics are the wavefunctions that describe the rotational motion of a rigid rotator, the names of the quantum numbers are changed to reflect the type of angular momentum encountered in the problem.