\n", " \n", "> [**Hamilton-Jacobi Equation**](https://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation) (classical mechanics)\n", ">\n", ">$$\n", "-\\frac{dS(x,t)}{dt} = H \n", "$$\n", "> where $H = T + V$ is the sum of the kinetic energy, $T$, and the potential energy, $V$, written a function of $S$, $x$, and $t$. In the simplest case, one has: \n", ">$$\n", "-\\frac{dS(x,t)}{dt} = \\left( \\frac{1}{2m} \\frac{dS(x,t)}{dx} \\cdot \\frac{dS(x,t)}{dx} + V(x,t) \\right)\n", "$$\n", "> because the momentum can be identified as $p(x,t) = \\tfrac{dS(x,t)}{dx} $.\n", "\n", "

\n", "\n", "> [**Schrödinger Equation**](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation) (quantum mechanics)\n", "> \n", ">$$ \\frac{i h}{2 \\pi} \\frac{d \\Psi(x,t)}{dt} = \\left( -\\frac{h^2}{2m (2 \\pi)^2 } \\frac{d^2 \\Psi(x,t)}{dx^2} + V(x,t)\\Psi(x,t) \\right) $$\n", ">The solution, $\\Psi(x,t)$ to the Schrödinger equation is called the [*wavefunction*](https://en.wikipedia.org/wiki/Wave_function). A key postulate of quantum mechanics is that the wavefunction contains all the information needed to fully specify the observable properties of a quantum system. \n", "\n", "

\n", "\n", "All three equations have strong resemblance. But, in particular, the Schrödinger equation and the Hamilton-Jacobi equation are consistent with each other. Suppose one uses the wavefunction ansatz,\n", "\n", "$$ \\Psi(x,t) = \\psi_0(x,t) e^{\\frac{2 \\pi i S(x,t)}{h}} $$\n", "\n", "where $\\psi_0(x,t)$ and $S(x,t)$ are, without loss of generality, assumed to be real-valued functions. We will insert this expression into the Schrödinger equation, but first it is useful to define the quantity \n", "\n", "$$ \n", "\\hbar = \\tfrac{h}{2 \\pi} \n", "$$\n", "\n", "One then can rewrite the wavefunction and the Schrödinger equation without the pesky factors of $2 \\pi$, respectively,\n", "\n", "$$ \\Psi(x,t) = \\psi_0(x,t) e^{\\frac{i S(x,t)}{\\hbar}} $$\n", "\n", "$$ i \\hbar \\frac{d \\Psi(x,t)}{dt} = \\left( -\\frac{\\hbar^2}{2m} \\frac{d^2 \\Psi(x,t)}{dx^2} + V(x,t)\\Psi(x,t) \\right) $$\n", "Inserting the wavefunction into the Schrödinger equation, the left-hand-side is\n", "\n", "$$\n", " i \\hbar \\frac{d \\Psi(x,t)}{dt} =e^{\\frac{i S(x,t)}{\\hbar}} \\left( i \\hbar \\frac{d \\psi_0(x,t)}{dt}-\\psi_0(x,t) \\frac{d S(x,t)}{dt} \\right)\n", "$$\n", "\n", "and the right-hand-side is\n", "\n", "$$\n", "\\begin{align}\n", "& \\left( -\\frac{\\hbar^2}{2m} \\frac{d^2 \\Psi(x,t)}{dx^2} +V(x,t)\\Psi(x,t) \\right) \\\\\\\\\n", "& = e^{\\frac{i S(x,t)}{\\hbar}} \\left(-\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi_0(x,t)}{dx^2} - \\frac{i \\hbar}{m}\\frac{d \\psi_0(x,t)}{dx}\\frac{d S(x,t)}{dx} - \\frac{i \\hbar}{2m} \\psi_0(x,t) \\frac{d^2 S(x,t)}{dx^2} + \\frac{1}{2m}\\psi_0(x,t)\\left(\\frac{d S(x,t)}{dx}\\right)^2 + V(x,t) \\psi_0(x,t) \\right) \n", "\\end{align}\n", "$$\n", "\n", "Inserting these equations into the Schrödinger equation and dividing through by $e^{\\frac{i S(x,t)}{\\hbar}}$ gives:\n", "\n", "$$\n", "\\begin{align} \n", "& \\left( i \\hbar \\frac{d \\psi_0(x,t)}{dt}-\\psi_0(x,t) \\frac{d S(x,t)}{dt} \\right) \\\\\\\\\n", "& = \\left(-\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi_0(x,t)}{dx^2} - \\frac{i \\hbar}{m}\\frac{d \\psi_0(x,t)}{dx}\\frac{d S(x,t)}{dx} - \\frac{i \\hbar}{2m} \\psi_0(x,t) \\frac{d^2 S(x,t)}{dx^2} + \\frac{1}{2m}\\psi_0(x,t)\\left(\\frac{d S(x,t)}{dx}\\right)^2 + V(x,t) \\psi_0(x,t) \\right) \n", "\\end{align}\n", "$$\n", "\n", "The real and imaginary parts of this equation must be separately equal to each other. The real part of the equation is \n", "\n", "$$\n", "\\begin{align}\n", "-\\psi_0(x,t) \\frac{d S(x,t)}{dt}\n", "= \\left(-\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi_0(x,t)}{dx^2} + \\frac{1}{2m}\\psi_0(x,t)\\left(\\frac{d S(x,t)}{dx}\\right)^2 + V(x,t) \\psi_0(x,t) \\right) \n", "\\end{align}\n", "$$\n", "\n", "Dividing both sides by $\\psi_0(x,t)$ and taking the classical limit, where $\\hbar \\rightarrow 0$, gives back the Hamilton-Jacobi equation,\n", "\n", "$$ - \\frac{d S(x,t)}{dt}\n", "= \\left(\\frac{1}{2m}\\left(\\frac{d S(x,t)}{dx}\\right)^2 + V(x,t) \\right) $$\n", "\n", "The imaginary part of the equation is:\n", "\n", "$$ \\hbar \\frac{d \\psi_0(x,t)}{dt} \n", "= - \\left(\\frac{\\hbar}{m}\\frac{d \\psi_0(x,t)}{dx}\\frac{d S(x,t)}{dx} + \\frac{\\hbar}{2m} \\psi_0(x,t) \\frac{d^2 S(x,t)}{dx^2} \\right) $$\n", "\n", "We notice that this expression is zero in the classical limit. Dividing through by $\\hbar$ and using the chain rule, $\\frac{d \\left(\\psi_0(x,t)\\right)^2}{dt} = 2 \\psi_0(x,t) \\frac{d \\psi_0(x,t)}{dt}$, we can rewrite this expression as\n", "\n", "$$ \\frac{d \\left(\\psi_0(x,t)\\right)^2}{dt} \n", "= - \\frac{d}{dx} \\left(\\psi_0(x,t)\\right)^2 \\left(\\frac{1}{m}\\frac{d S(x,t)}{dx}\\right) $$\n", "\n", "This expression resembles the [*equation of continuity*](https://en.wikipedia.org/wiki/Continuity_equation) from classical physics,\n", "\n", "$$ \\frac{d \\rho(x,t)}{dt} = -\\nabla \\cdot \\rho(x,t) \\nabla v(x,t) $$\n", "\n", "where $v(x,t)$ is the velocity and $\\rho(x,t)$ is the density, the probability of observing a particle at the point $x$ and the time $t$. This (re)confirms that $\\nabla S$ is an expression for the momentum, and suggests that\n", "\n", "$$\\rho(x,t) = \\psi_0(x,t)^2 = \\Psi(x,t) \\Psi(x,t)^* = |\\Psi(x,t)|^2 $$ \n", "\n", "is an expression for the probability of observing a particle at a point in space. The identification of the square-magnitude of the quantum-mechanical wavefunction with probability is called the [*Born postulate*](https://en.wikipedia.org/wiki/Born_rule).\n", "\n", "\n", "In the end, the Schrödinger equation was not accepted because of its mathematical elegance, but because it worked to accurately describe key phenomena. In fact, as we shall see, the (originally empirical) Rydberg formula can be easily justified using the Schrödinger equation.\n", "\n", "We can rationalize the form of the quantum mechanical momentum operator, $\\hat{p} = -i \\hbar \\frac{d}{dx} $\n", ", by a similar argument. Applying the hypothesized momentum operator to the wavefunction, one obtains:\n", "\n", "$$\n", "\\begin{align}\n", "-i \\hbar \\frac{d\\Psi(x,t)}{dx} &= -i \\hbar \\frac{d\\psi_0(x,t)}{dx} e^{\\frac{i S(x,t)}{\\hbar}} + \\psi_0(x,t) e^{\\frac{i S(x,t)}{\\hbar}} \\frac{dS(x,t)}{dx} \\\\\\\\\n", "&= \\text{[imaginary part]} + \\frac{dS(x,t)}{dx} \\Psi(x,t) \\\\\\\\\n", "&= \\text{[imaginary part]} + \\text{[classical expression for momentum]}\\Psi(x,t)\n", "\\end{align}\n", "$$\n", "\n", "This indicates that the quantum-mechanical momentum operator is consistent with the classical expression for the momentum, to which it reduces in the classical limit where $\\hbar \\rightarrow 0$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 🪞 Self-Reflection\n", "Imagine you were assigned to describe Quantum Mechanics to a room full of irrascible tweens. \n", "- How would you motivate them to be interested in quantum mechanics? \n", "- How would you describe wave-particle duality in a way they could understand? " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ❓ Knowledge Test\n", "- This [set of questions](IntroTestExample.ipynb) demonstrates the different types of knowledge-test questions that appear in the course, and how to solve them. [GitHub Classroom Link](https://classroom.github.com/a/d3E1lA4j)\n", "- More questions, mostly about wave-particle duality. [GitHub Classroom Link](https://classroom.github.com/a/DMg3v70o)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 🤔 Thought-Provoking Questions\n", "- Is it possible for the wavelength of the scattered photon in Compton Scattering to be infinite? (I.e., could the photon be entirely absorbed?)\n", "- What are the units of the wavefunction?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 🔁 Recapitulation\n", "- What are the following phenomena/experiments/theories, and why were they important to the establishment of the quantum theory?\n", " - black-body radiation and the ultraviolet catastrophe\n", " - the photoelectric effect\n", " - Compton scattering\n", " - the Davisson-Germer experiment (and electron scattering/diffraction in general)\n", " - the spectral lines of one-electron atoms, and the Rydberg relation\n", "- What are the following key equations, and why are they important?\n", " - Planck-Einstein relation for the energy of a photon\n", " - De Broglie relation between momentum and wavelength\n", " - the Schrödinger equation\n", "- What is the quantum mechanical operator for the momentum of a particle?\n", "- List as many properties/characteristics of the wavefunction as you can." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 🔮 Next Up...\n", "- Motivate the time-dependent Schrödinger equation\n", "- Derive the time-independent Schrödinger equation\n", "- Explore the quantum-mechanical operator for momentum\n", "- Study simple one-dimensional Hamiltonians" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 📚 References\n", "My favorite sources for this material are:\n", "- R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, New York, 1974)\n", "- R. Dumont, [An Emergent Reality, Part 2: Quantum Mechanics](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf) (Chapters 1 and 2).\n", "- Also see my (pdf) class [notes](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/IntroQM.pdf).\n", "- The introductory material in [Davit Potoyan's](https://www.chem.iastate.edu/people/davit-potoyan) Jupyter-book course is very good. Roughly [chapters 1 and 2](https://dpotoyan.github.io/Chem324/intro.html) are especially relevant here.\n", "\n", "Some videos:\n", "- [My video introduction](https://www.macvideo.ca/media/1_IntroQM/1_rbe31g5b)\n", "- [History of Black-body radiation](https://www.youtube.com/watch?v=uMUi3o78qgQ&t=8s)\n", "- [History of the Photoelectric effect](https://www.youtube.com/watch?v=BiPEY99w8Lo)\n", "\n", "There are also some excellent wikipedia articles:\n", "- On General Quantum Mechanics\n", " - [Introduction to Quantum Mechanics](http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics)\n", " - [Quantum Mechanics](http://en.wikipedia.org/wiki/Quantum_mechanics)\n", " - [Mathematical Formulation of Quantum Mechanics](http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics)\n", " - [Basic Concepts of Quantum Mechanics](http://en.wikipedia.org/wiki/Basic_concepts_of_quantum_mechanics). See [also](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/quantum_summary.pdf)\n", " - [Planck's Constant](http://en.wikipedia.org/wiki/Planck_constant)\n", " - [Wave-particle Duality](http://en.wikipedia.org/wiki/Wave-particle_duality)\n", "- Black-Body Radiation\n", " - [What is a Black Body?](http://en.wikipedia.org/wiki/Black_body)\n", " - [Ultraviolet Catastrophe](http://en.wikipedia.org/wiki/Ultraviolet_catastrophe)\n", " - [Raleigh-Jeans Law for Classical Black Bodies](http://en.wikipedia.org/wiki/Rayleigh-Jeans_law)\n", " - [Planck's Law for Black-Body Radiation](http://en.wikipedia.org/wiki/Planck%27s_law)\n", " - [Wien's Displacement Law for the Peak Wavelength of a Black Body](http://en.wikipedia.org/wiki/Wien%27s_displacement_law)\n", "- [Photoelectric Effect](http://en.wikipedia.org/wiki/Photoelectric_effect)\n", "- [Compton Scattering](http://en.wikipedia.org/wiki/Compton_scattering)\n", "- [De Broglie Wavelength](http://en.wikipedia.org/wiki/Matter_wave)\n", "- Electron/Particle Diffraction\n", " - [Davisson-Germer Experiment](http://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment)\n", " - [Electron Diffraction](http://en.wikipedia.org/wiki/Electron_diffraction)\n", " - [Electron Crystallography](http://en.wikipedia.org/wiki/Electron_crystallography)\n", " - [Electron Microscope](http://en.wikipedia.org/wiki/Electron_microscopy)\n", " - [Transmission Electron Microscope](http://en.wikipedia.org/wiki/Transmission_Electron_Microscopy)\n", "- Spectral lines in simple atoms\n", " - [Franck-Hertz Experiment, the first experiment to show discrete atomic energy levels](http://en.wikipedia.org/wiki/Franck%E2%80%93Hertz_experiment)\n", " - [Balmer formula, a predecessor of the Rydberg formula](http://en.wikipedia.org/wiki/Balmer_series)\n", " - [Rydberg formula](http://en.wikipedia.org/wiki/Rydberg_formula)\n", " - [Spectral Lines](http://en.wikipedia.org/wiki/Spectral_line)\n", " - [Atomic Spectral Lines](http://en.wikipedia.org/wiki/Atomic_spectral_line)\n", "- Schrödinger Equation\n", " - [Schrödinger Equation](http://en.wikipedia.org/wiki/Schrodinger_equation)\n", " - [Theoretical Justification of S.E.](http://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation). See [also](https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/SchrodingerHJeq.pdf).\n", " - [Momentum Operator](http://en.wikipedia.org/wiki/Momentum_operator)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": { "height": "688px", "width": "378px" }, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": { "height": "602.857px", "left": "426px", "top": "183.571px", "width": "236.708px" }, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 4 }