Quantum Mechanics

a_brief_history_of_quantum_mechanics

Anonymous (anonymous@undisclosed.example.com) — 2021-01-29T06:51:00+00:00

In this module, we will study the phenomena that led to the development of quantum mechanics, in roughly chronological order. This is to help you understand where the formalism of quantum mechanics comes from. Despite their importance in the development of quantum mechanics, many of the phenomena described here will not be rederived from quantum mechanics later in the course because their full treatment is quite advanced, e.g., requiring a fully quantum treatment of the electromagnetic field. …

adjoints_and_hermitian_operators

Anonymous (anonymous@undisclosed.example.com) — 2022-10-06T00:45:36+00:00

Adjoints The Hermitian adjoint $\hat{A}^{\dagger}$ of an operator $\hat{A}$ is the unique operator such that \[\sand{\psi}{\hat{A}^{\dagger}}{\phi} = \sand{\phi}{\hat{A}}{\psi}^*,\] for all vectors $\ket{\psi}$ and $\ket{\psi}$. Note that multiplication by a scalar $\hat{a}\ket{\psi} = a\ket{\psi}$ can be thought of as a linear operator. Its Hermitian adjoint can be found via \begin{align*} \sand{\psi}{\hat{a}^{\dagger}}{\phi} & = \sand{\phi}{\hat{a}}{\psi}^* \\ & = \left ( a \braket{\phi}{\p…

atomic_transitions_and_spectroscopy

Anonymous (anonymous@undisclosed.example.com) — 2022-09-06T18:23:50+00:00

1.viii.1 Problems with the Classical Model of Atoms In 1911, Ernst Rutherford discovered the atomic nucleus and posited that atoms consist of negatively charged electrons orbiting a positively charged nucleus under the electrostatic interaction, in a similar way to how planets orbit the sun under the gravitational interaction.$10^{-8}\,\text{s}$$E_1$$E_2$$E_3$$\cdots$$\hbar = h/2\pi$\[L = n\hbar,\qquad\qquad n=1,2,3,\cdots,\]$E_m$$E_n$$E_n > E_m$$\nu$\[h\nu = E_n - E_m.\]$h\nu$$E_n$$E_m$$\nu$\[…

basic_properties_of_linear_operators

Anonymous (anonymous@undisclosed.example.com) — 2022-09-27T18:38:24+00:00

A linear operator $\hat{A}$ on a vector space is a function that maps vectors to vectors, $|\psi \rangle \rightarrow \hat{A}|\psi\rangle$ such that \[\hat{A} \left ( a |\psi \rangle + b|\phi \rangle \right ) = a\hat{A} |\psi\rangle + b\hat{A} |\phi \rangle .\] Given a linear operator $\hat{A}$ on an inner product space, we can also define an action of a linear operator on a dual vector. The dual vector $\langle \psi | \hat{A}$ is defined to be the unique dual vector such that \[\left ( \langle…

blackbody_radiation

Anonymous (anonymous@undisclosed.example.com) — 2021-01-29T23:06:49+00:00

1.ii.1 What is a Blackbody? Heating metal causes it to radiate heat and light. The color/frequency of radiation changes as we increase temperature. The following video shows how the color of the radiation changes as a block of iron is heated up. When current is passed through the heating element it heats up, but also becomes magnetized like a solenoid because it is a coil. A block of magnetized iron is used that floats above the heating element so that you can see its color more clearly, but…

commutators_and_uncertainty_relations

Anonymous (anonymous@undisclosed.example.com) — 2022-10-06T01:02:30+00:00

Commutators The commutator of two operators $\hat{A}$ and $\hat{B}$ is \[\boxed{[\hat{A},\hat{B} ] = \hat{A}\hat{B}-\hat{B}\hat{A}.}\] Two operators commute if \[[\hat{A},\hat{B}] = 0,\] or, equivalently \[\hat{A}\hat{B} = \hat{B}\hat{A}.\] Clearly, an operator always commutes with itself $[\hat{A},\hat{A}] = 0$. A little less obviously, if $\hat{A}$, $\hat{B}$ and $\hat{A}\hat{B}$ are all Hermitian then $\hat{A}$ and $\hat{B}$ commute. Here is the proof: \begin{align*} [\hat{A},\hat{B}] & …

continuous_basis_representations

Anonymous (anonymous@undisclosed.example.com) — 2021-03-12T21:25:55+00:00

Consider a continuously infinite dimensional Hilbert space, i.e. it has a basis $\ket{\chi_k}$ labelled by a continuous index $k$. We call such a basis orthonormal if \[\braket{\chi_{k'}}{\chi_k} = \delta(k-k').\] Note that this means that \[\braket{\chi_k}{\chi_k} = \delta(0),\] and by the integral representation of the Dirac delta function this is \[\delta(0) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\D k\, e^{i0} = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\D k\,=\infty,\]$\|\chi_k\| \neq 1$\[\ha…

de_broglie_matter_waves

Anonymous (anonymous@undisclosed.example.com) — 2022-10-13T18:00:03+00:00

1.v.1 de Broglie Wavelength and Wave Vector The photoelectric effect and Compton scattering show that electromagnetic waves sometimes exhibit particle-like properties. In 1923, Louis de Broglie proposed that matter, which we normally think of as made up of particles like electrons, protons and neutrons, should have wave-like properties. In other words, $E = pc$$E = h\nu$$p = h\nu /c = h/\lambda$$\lambda = h/p$$p$\[\boxed{\lambda = \frac{h}{p}.}\]$\vec{p}$$\vec{k}$$k = \frac{2\pi}{\lambda}$\…

dirac_notation

Anonymous (anonymous@undisclosed.example.com) — 2021-02-22T21:15:38+00:00

In quantum mechanics, we usually use Dirac Notation for Hilbert spaces. The advantages of Dirac notation are: It allows us to work in a basis independent notation.We can use the same notation for both finite and infinite dimensional Hilbert spaces.The correspondence between acting with a dual vector on a vector and an inner product $f_{\phi}(\psi) = (\phi,\psi)$$\psi$$|\psi\rangle$$f_{\psi}$$\psi$$f_{\psi}$$f_{\psi}(\phi) = (\psi,\phi)$$\phi$$\langle \psi |$$(\phi,\psi)$$\langle \phi | \psi \ra…

eigenvalues_and_eigenvectors

Anonymous (anonymous@undisclosed.example.com) — 2022-10-06T23:47:11+00:00

A vector $\ket{\psi}$ is called an eigenvector (also called an eigenstate in quantum mechanics) of an operator $\hat{A}$ if \[\hat{A} \ket{\psi} = a \ket{\psi},\] where $a$ is a scalar called an eigenvalue of $\hat{A}$. As an example, all vectors are eigenvectors of the identity operator $\hat{I}$ with eigenvalue $1$, since \[\hat{I} \ket{\psi} = \ket{\psi} = 1\ket{\psi}.\] The importance of eigenvalues and eigenvectors is that $\ket{a}$$\hat{A}$$a$$\sand{a}{\hat{A}}{a}$$\hat{A}$\[\sand{a}{\ha…

fourier_series_and_fourier_transforms

Anonymous (anonymous@undisclosed.example.com) — 2021-03-10T22:05:15+00:00

2.v.1 Fourier Series A periodic function $f$ with period $L$ (subject to certain continuity conditions that I will not detail here) can be written as \[\boxed{f(x) = a + \sum_{n=1}^{\infty} b_n \cos \left ( \frac{2 \pi n x}{L} \right )+ \sum_{n=1}^{\infty} c_n \sin \left ( \frac{2\pi n x}{L}\right ).}\] This is called the Fourier Series for the function $f$. The figure below shows how a square wave can be decomposed into a sum of sinusoidal terms, giving a closer approximation as higher frequ…

functions_inverses_and_unitary_operators

Anonymous (anonymous@undisclosed.example.com) — 2022-10-06T01:05:47+00:00

Functions of Operators Let $f$ be a (complex) function with Taylor expansion \[f(z) = \sum_{n=0}^{\infty} a_n z^n,\] and radius of convergence $|z| \leq r$. We can extend $f$ to be a function on linear operators by defining \[f(\hat{A}) = \sum_{n=0}^{\infty} a_n \hat{A}^n.\] It is possible to prove that this series converges if \[\sup_{\{\ket{\psi}| \| \psi \| = 1 \}} \Abs{\sand{\psi}{\hat{A}}{\psi}} \leq r.\] Interaction of Functions with Commutators $[\hat{A}+\hat{B},\hat{C}] = [\hat{A},\…

general_considerations

Anonymous (anonymous@undisclosed.example.com) — 2020-07-21T05:39:12+00:00

4.iii.1 Introduction This section is about general features of the solutions to the one-dimensional Time Independent Schrödinger Equation (TISE) \[-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi(x)}{\mathrm{d}x^2} + V(x)\psi(x) = E\psi(x)\] Recall that the TISE is an eigenvalue equation $\hat{H}|\psi\rangle=E|\psi\rangle$ with the Hamiltonian, written in the position representation \[\hat{H}\rightarrow -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x} + V(x)\]$E$$\hat{H}$$Ae^{ipx/\hbar}$$\psi(x) = …

hilbert_space

Anonymous (anonymous@undisclosed.example.com) — 2021-04-02T19:03:40+00:00

The mathematical formalism of quantum mechanics is expressed in Hilbert space. The wavefunctions $\psi(x,t)$ that we have seen so far are elements of an infinite-dimensional vector space with a continuous basis, with additional properties needed to ensure that probabilities are well defined.$\psi,\phi,\chi ,\cdots$$a,b,c,\cdots$$\psi + \phi$$a \psi$$\psi$$\phi$$\psi + \phi$$\psi + \phi = \phi + \psi$$(\psi + \phi) + \chi = \psi + (\phi + \chi)$$\boldsymbol{0}$$\psi$\[\boldsymbol{0} + \psi = \ps…

mathjax

Anonymous (anonymous@undisclosed.example.com) — 2021-02-23T07:19:17+00:00

MathJax.Hub.Config({ TeX: { Macros: { D: "{\\mathrm{d}}", ket: ["{\\left \\lvert #1 \\right \\rangle}",1], bra: ["{\\left \\langle #1 \\right \\rvert}",1], braket: ["{\\left \\langle #1 \\middle \\vert #2 \\right \\rangle }",2], sand: ["{\\left \\langle #1 \\middle \\vert #2 \\middle \\vert #3 \\right \\rangle}",3], ketbra: ["{\\left \\lvert #1 \\middle \\rangle \\middle \\langle #2 \\right \\rvert}",2], proj: ["{\\ketbra{#1}{#1}}",1], Abs: ["{\\left \\lvert #1 …

module_1_summary

Anonymous (anonymous@undisclosed.example.com) — 2021-03-01T18:35:12+00:00

Blackbody Radiation Planck derived the blackbody spectrum by assuming that matter and radiation can only exchange energy in discrete chunks (quanta): \[E(\nu) = nh\nu \qquad \text{for} \qquad n=0,1,2,3,\cdots\]He obtained: \[\boxed{u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1},}\]To fit experimental data, he needed a new constant called Planck's Constant with value \[h = 6.626\times 10^{-34}\,\text{m}^2\text{kg}\,\text{s}^{-1}.\]\[\boxed{I = a\sigma T^4},\]$a=1$$a<1$$\sigma = 5.67\time…

probability_and_uncertainty_in_quantum_mechanics

Anonymous (anonymous@undisclosed.example.com) — 2022-09-06T18:17:37+00:00

1.vii.1 The Nature of Probability in Quantum Physics Many physicists are inclined to say that there must be a fundamental indeterminism in physics. After all, if we cannot observe an electron's trajectory when it is involved in forming an interference pattern without ruining the interference patter then doesn't that mean that the electron does not $\psi(x,t)$\[\int_{-\infty}^{+\infty} |\psi(x,t)|^2\,\mathrm{d}x = 1.\]$t$$x$$a

start

Anonymous (anonymous@undisclosed.example.com) — 2021-01-29T06:20:31+00:00

Origins of Quantum MechanicsA Brief History of Quantum MechanicsBlackbody RadiationThe Photoelectric EffectThe Compton Effectde Broglie Matter WavesWave-Particle DualityProbability and Uncertainty in Quantum MechanicsAtomic Transitions and SpectroscopyWave PacketsSummaryMathematical ToolsHilbert SpaceDirac NotationLinear OperatorsBasic Properties of Linear OperatorsAdjoints and Hermitian OperatorsCommutators and Uncertainty RelationsFunctions, Inverses, and Unitary OperatorsEigenvalues and Eigen…

the_compton_effect

Anonymous (anonymous@undisclosed.example.com) — 2022-10-13T17:36:39+00:00

In 1923, Arthur Compton found that the wavelength of X-rays scattered off free electrons was larger than the wavelength of the incident radiation. Classically, X-rays are too energetic to be absorbed by free electrons. They would provide an oscillating electromagnetic field that would cause the electrons to oscillate at the same frequency. Therefore, they should emit X-rays with the same frequency as the incident radiation.$E=h\nu$\[\vec{p} = \vec{p} + \vec{P}_e \qquad\qquad \Rightarrow \qqua…

the_delta_function_potential

Anonymous (anonymous@undisclosed.example.com) — 2020-07-20T23:52:54+00:00

In this section, we study the attractive delta function potential: \[V(x) = -\alpha \delta(x),\] where $\alpha$ is a positive constant. Now, obviously a potential that dips down to $-\infty$ at just a single point is highly idealized. However, solving this system is a good stepping stone from the infinite square well to the finite square well.$E < 0$$E > 0$$E < 0$$x=0$$E>0$$x < 0$$E$$x=0$$x=0$$E$$x>0$$x < 0$$x > 0$$x=0$$E$$k = \sqrt{-\frac{2mE}{\hbar^2}}$\[\psi(x) = \begin{cases} \psi_-(x) = A…

the_photoelectric_effect

Anonymous (anonymous@undisclosed.example.com) — 2022-09-01T20:11:04+00:00

1.iii The Photoelectric Effect When we shine light on a metal, electrons are emitted. The following facts were observed prior to 1905: If the frequency of radiation is below a cutoff frequency $\nu_0$ (where $\nu_0$ depends on the material) then no electrons are emitted.$\nu_0$$\nu$\[\boxed{E = h\nu,}\]$W$$K$\[h\nu = W + K.\]$h\nu$$W$$\nu_0$$h\nu_0 = W$$\nu_0 = W/h$\[K = h(\nu - \nu_0).\]$\nu$$K$$V$$\nu$$V$$V_s$$h$$e$$\nu$$W$

the_schroedinger_equation

Anonymous (anonymous@undisclosed.example.com) — 2021-03-17T21:25:16+00:00

The second postulate of quantum mechanics is: Dynamics: The state of a closed quantum system (i.e. one that is not being measured or interacting with its environment) evolves in time according to the Schrödinger equation: \[i\hbar \frac{\partial \ket{\psi(t)}}{\partial t} = \hat{H}\ket{\psi(t)},\] where $\hat{H}$ is the Hamiltonian operator. $\psi(x,t) = Ae^{i(kx-\omega t)}$$p = \hbar k$$E = \hbar \omega$\[\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 \psi}{\partial t^2},\…

the_spectral_theorem

Anonymous (anonymous@undisclosed.example.com) — 2022-10-08T00:03:59+00:00

Proof of the Spectral Theorem for Normal Operators in Finite Dimensions As a reminder, a normal operator is an operator $\hat{A}$ that satisfies $[\hat{A},\hat{A}^{\dagger}] = 0$, and the spectral theorem is as follows. Theorem Let $a_1,a_2,\cdots$ be the (distinct) eigenvalues of an operator $\hat{A}$. Then, $\hat{A}$ is a normal operator if an only if all of the following hold:$\hat{P}_{a_j}\hat{P}_{a_k} = \delta_{jk}\hat{P}_{a_j}$$\sum_j \hat{P}_{a_j} = \hat{I}$$\hat{A} = \sum_j a_j \hat{P…

the_variational_principle

Anonymous (anonymous@undisclosed.example.com) — 2020-08-24T07:02:30+00:00

5.i.1 Introduction This section discusses two related concepts: the variational principle and the variational method (also called the Rayleigh-Ritz method). The variational principle shows that solving the Time Independent Schrödinger Equation (TISE) is equivalent to the stationarity of the energy functional $E[\psi]$$\delta S$$\hat{H}$\[ E[ \psi] = \langle \hat{H} \rangle = \langle \psi | \hat{H} | \psi \rangle ,\]$|\psi\rangle$$x$$\psi(x)$$E[\psi]$$|\delta \psi \rangle$\[\delta E [\psi] = E[…

wave_packets

Anonymous (anonymous@undisclosed.example.com) — 2021-02-18T00:24:36+00:00

1.ix.1 Wave Packets and Fourier Transforms A wavefunction of the form $\psi(x,t) \propto e^{i(kx - \omega t)}$ has a well-defined momentum, but is spread out over all space. See the graph below for an illustration of what this wavefunction looks like for $k = \pi$, $t=0$. The wavefunction $\psi(x,t) \propto e^{i(kx - \omega t)}$ is also not normalizable because \[\int_{-\infty}^{+\infty} |\psi(x)|^2\,\mathrm{d}x = \int_{-\infty}^{+\infty} \,\mathrm{d}x = \infty,\]$x = x_0$$\psi(x,t) = \delta…

wave-particle_duality

Anonymous (anonymous@undisclosed.example.com) — 2022-09-06T18:08:18+00:00

1.vi.1 Particles vs. Waves In classical physics, particles and waves are mutually exclusive. A particle is described by its position $\vec{r}(t)$ and momentum $\vec{p}(t)$, and always has a well-defined trajectory in space-time.A wave is an excitation of a field and is specified by its wavefunction, e.g. $\psi(\vec{r},t) = Ae^{i(\vec{k}\cdot\vec{r} - \omega t)}$$I_1$$I_2$$I$$I = I_1 + I_2$$|\psi(\vec{r},t)|^2$$\psi(\vec{r},t)$$\psi(\vec{r},t)$$\psi_1(\vec{r},t)$$\vec{r}$$\psi_2(\vec{r},t)$$\psi…