http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-27T04:03:36-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2011-09-05T12:31:31-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spleccommutation?rev=1315251091 Commutation and Uncertainty (Lecture, 20 minutes) Pages 34-39 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spleccommutators?rev=1313543066 Commutators & Commuting Operators (10 minutes) Pages 34-37 $$\left[S_{x},S_{y}\right]=S_{x}S_{y}-S_{y}S_{x} \; \; .$$ “” $$\left[S_{x},S_{y}\right]=i\hbar S_{z} \; \; , $$ $$\left[S_{y},S_{z}\right]=i\hbar S_{x} \; \; , $$ $$\left[S_{z},S_{x}\right]=i\hbar S_{y} \; \; . $$ text/html 2017-01-31T13:35:25-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spleccomprel?rev=1485898525 The Completeness Relation (10 minutes) Page 14 $$\vert +\rangle \langle + \vert \; + \; \vert- \rangle\langle -\vert=1 \; \; . $$ Be sure to note that this can be done in any basis, but you have to pick one and stick to it to satisfy the completeness relation. text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spleccorrespon?rev=1313543066 The Correspondence Principle (5 minutes) (Slide 5) text/html 2015-08-15T13:46:35-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecdensop?rev=1439671595 Introducing the Density Operator (20 minutes) “” $$P_{+} \, = \, \left\vert\langle +\vert\psi \rangle\right\vert^{2}=\langle \psi\vert+ \rangle\langle +\vert\psi \rangle \; \; .$$ So, the rightmost expression is some number its complex conjugate. But, we don't have to think of the expression this way. Notice that in the middle there's a projection operator hidden between the $\psi$ terms. text/html 2011-07-20T15:24:31-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecdensopapp?rev=1311200671 Applications for the Density Function (10 minutes) This lecture is best performed following the Describing the Sample Oven in the Stern-Gerlach Experiment Mathematically (FIXME Needs link) activity. $$\vert+ \rangle_{y}{_{y}\langle +\vert}\psi \rangle\langle \psi\vert \; \; .$$ text/html 2015-08-15T13:47:00-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecdfindspinupx?rev=1439671620 Deriving Spin-up with x-direction in the z-basis (25 minutes) $$\vert+ \rangle_{x}=a \vert+ \rangle + b\vert- \rangle \; \; .$$ Note that the white arrows represent $|+ \rangle$ and $|- \rangle$ while the green arrows represent $\vert+ \rangle_{x}$ and $\vert- \rangle_{x}$. However, we have no more room in our two-dimensional vector space to properly draw $\vert+ \rangle_{y}$ and $\vert- \rangle_{y}$ in a manner that reflects the results of the Stern-Gerlach experiment and the orthogona… text/html 2011-07-20T10:57:35-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecdipolemoment?rev=1311184655 The Magnetic Dipole Moment (10 minutes) Lorentz Force and Work on a Rectangular Loop $$U=- \vec{\mu} \cdot \vec{B}$$ can be written on the board. “” text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecehrenfest?rev=1313543066 Ehrenfest's Theorem (10 minutes) (Slides 1-3) text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spleceigenstate?rev=1313543066 Energy eigenvalues and eigenstates (Lecture, 60 minutes) Pages 6-11 <http://phet.colorado.edu/en/simulation/bound-states> text/html 2012-01-25T10:51:37-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecexpectationvalue?rev=1327517497 Expectation Value and Standard Deviation(Lecture, 20 minutes) Pages 26-33 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecexpecval?rev=1313543066 Expectation Values & Quantum Uncertainty (15 minutes) Pages 29-33, 38 Completeness Relation $$\frac{+ \hbar}{2}\langle \psi \vert +\rangle\langle+ \vert\psi \rangle \; \; + \; \; \frac{- \hbar}{2}\langle\psi \vert- \rangle\langle -\vert\psi \rangle \; \; . $$ text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecgeneralizedspin?rev=1313543066 Spin N systems (Lecture, 10 minutes) Pages 26-27 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splechistoricsg?rev=1313543066 Discussion of Historic Stern-Gerlach Experiment (Lecture, 20 minutes) Pages 5-11 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecintrinsicspin?rev=1313543066 Discussion of Intrinsic Spin (Lecture, 15 minutes) Pages 10-11 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecintro?rev=1313543066 Introduction to the Spins Course (Lecture, 15 minutes) Pages 1-4 text/html 2012-02-13T15:16:43-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecintroprob?rev=1329175003 Introduction to Probability ( minutes) Handout: [spspin1guide.pdf] (pages 4-8) text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecintrospinsprog?rev=1313543066 Introduction to the SPINS Program (Lecture, 10 minutes) Page 12 text/html 2011-07-20T16:23:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecketdimension?rev=1311204192 Dimensionality of the Ket Vector Space (10 minutes) $$\mathcal{P}_{+}=\left\vert\langle+\vert-\rangle\right\vert^{2}$$ and $$\mathcal{P}_{-}=\left\vert\langle-\vert-\rangle\right\vert^{2} \; \; , $$ respectively. text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecmagres?rev=1313543066 Magnetic Resonance (40 minutes) (Slides 9-16) Providing the slides in a handout form is often beneficial for students so they can discuss and review the calculations at a later time and at their own pace. text/html 2011-07-22T11:40:49-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecmeasure?rev=1311360049 How Making Measurements Affect Results in Quantum Systems (5 minutes) Classically:onlyIt appears that the particles running through the sorters “forgot” whether they were long or short.A device that recombines two beams in this way is called an interferometer.The socks now remember if they were long or short after the beam is recombined.“” Be sure to make the correspondence between the quantum mechanical case with particles and the sorting of socks clear throughout the exercise. No… text/html 2011-07-20T14:31:54-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecnormalize?rev=1311197514 Normalization of Quantum State Vectors (5 minutes) $$\vert\psi \rangle=ae^{i \alpha}\vert+ \rangle + be^{i \beta}\vert- \rangle \; \; , $$ the quantum state vector is normalized if $$\left\vert ae^{i \alpha}\right\vert^{2} + \left\vert be^{i \beta}\right\vert^{2}=1 \; \; ,$$ text/html 2011-09-05T11:47:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecoperators?rev=1315248453 Operators (Lecture, 60 minutes) Pages 2-12 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecoscillation?rev=1313543066 Neutrino Oscillations (Lecture, XX minutes) Pages 36-42 a bit beyond what our students were ready to hear, having no exposure to particle physics at this stage text/html 2015-08-15T13:46:52-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecouterproduct?rev=1439671612 The Outer Product (10 minutes) is not So, this operation allows us to move from the bra-ket notation of the projection operator $P_{\alpha}$ and find our matrix representation. Compute Several Outer Products text/html 2012-07-08T14:50:31-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecpostulate4?rev=1341784231 Postulate 4 of Quantum Mechanics (10 minutes) “The probability of obtaining $+ \frac{\hbar}{2} $ in a measurement of the observable $S_{z}$ on a system in the state $\vert \psi \rangle $ is $\mathcal{P}_{+}=\left\vert \langle +\vert \psi \rangle \right\vert^{2}$.” text/html 2017-01-31T13:26:18-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecpostulate5?rev=1485897978 The Projection Postulate of Quantum Mechanics (5 minutes) “After a measurement of some observable $\hat{A}$ that yields the result $a_{n}$, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement.” text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecprecession?rev=1313543066 Spin Precession (Lecture, XX minutes) Pages 12-35 text/html 2011-07-20T14:43:09-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecprojcollapse?rev=1311198189 The Projection Operator & Wave State Collapse (30 minutes) $$\vert\psi ' \rangle=\frac {P_{\vert\psi ' \rangle} \vert\psi \rangle}{\sqrt{\langle \psi|P_{\vert\psi ' \rangle}\vert\psi \rangle}} \; \; . $$ Unfortunately, there's no mathematically nice way to do this operation; to find the projection, the only choice is to plug in what we know and grind through computing the bra-kets. text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecprojoperator?rev=1313543066 The Projection Operator (10 minutes) Pages 14-15 ... text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecquanpostulates?rev=1313543066 Postulates of Quantum Mechanics (10 minutes) Pages 3-5 “” text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecrabiflop?rev=1313543066 Rabi Flopping (40 minutes) (Slides 1-8) $$\vert \langle out\vert in \rangle \vert ^{2} \; = \; ? $$ Providing the slides in a handout form is often beneficial for students so they can discuss and review the calculations at a later time and at their own pace. text/html 2011-07-20T14:29:47-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecrelphases?rev=1311197387 Relative Phases in Quantum States (10 minutes) text/html 2011-07-20T13:55:24-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecrepstates?rev=1311195324 Bra-ket Notation (5 minutes) Several Common Representations to Write on the Board $\vert \: + \rangle $ $\vert\: \frac{\hbar}{2} \rangle$ $\vert\: up \rangle$ $\vert\: \uparrow \rangle$ text/html 2016-01-18T13:49:47-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecringdipolemoment?rev=1453153787 Lecture ( minutes) text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecschrodinger?rev=1313543066 Schrodinger Equation (Lecture, 60 minutes) Pages 1-5 ... text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecsgsoftware?rev=1313543066 Introduction to the Stern Gerlach simulation (Lecture, 10 minutes) Page 12 text/html 2011-07-14T15:09:17-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecsgterm?rev=1310681357 Terminology for the Stern-Gerlach Experiment (5 minutes) “”“” text/html 2011-09-05T12:20:50-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecspin1?rev=1315250450 Spin 1 Systems (Lecture, 10 minutes) Pages 22,23 text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecspinprec?rev=1313543066 Spin Precession (5 minutes) (Slide 4) Ehrenfest's Theorem“”“” Providing the slides in a handout form is often beneficial for students so they can discuss and review the calculations at a later time and at their own pace. text/html 2011-09-05T13:17:51-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecspinprojection?rev=1315253871 , pages 13-21, 24, 40-41 Lecture (60 minutes) ->-> text/html 2011-09-05T12:13:45-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecspinvectors?rev=1315250025 The $S$ and $S^2$ vectors (Lecture, 15 minutes) Pages 42-45 text/html 2015-08-15T13:46:49-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecssquared?rev=1439671609 The $S^{2}$ Operator (15 minutes) Computing the $S^{2}$ Operator for the Spin-$\frac{1}{2}$ System text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecstateformalism?rev=1313543066 Building Quantum State Formalism (Lecture, 50 minutes) Page 18-25, 28 text/html 2012-07-09T15:10:13-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecstatisticchoice?rev=1341871813 How Choosing the # Trials vs. # Experiments Changes Results (15 minutes) probabilities of a spin-$\frac{1}{2}$ system ---------- Sample Data $x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$x_{7}$$x_{8}$$x_{9}$$x_{10}$# Spin-up particles7558578278 M=Number of events in a single experiment text/html 2014-12-31T16:23:23-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecstatisticchoice2?rev=1420071803 Lecture ( minutes) FIXME text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecstresults?rev=1313543066 Expected vs. Observed Results of the Stern Gerlach Experiment (10 minutes) Pages 5-8 $$U=\, -\vec{\mu} \cdot \vec{B} \; \; ,$$ text/html 2011-08-16T18:04:26-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:splecsuccessivesg?rev=1313543066 Outcomes using successive Stern Gerlach devices (Lecture, 40 minutes) Page 12-19 text/html 2012-08-09T09:06:08-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:spschrodinger?rev=1344528368 The Schrodinger Equation (25 minutes) $$i\, \hbar \, \frac{d}{dt}\, \vert \psi (t)\rangle =\hat{H} \, \vert\psi (t)\rangle \; \; . $$ $$\hat{H}\vert E_{i} \rangle=E_{i}\vert E_{i} \rangle \; \; .$$ $$\vert\psi (t)\rangle \: = \: \sum_{n}c_{n}\vert E_{n} \rangle \; \; .$$ text/html 2011-07-20T14:48:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:splec:the_hamiltonian?rev=1311198482 The Hamiltonian (10 minutes) “” $$H=- \: \vec{\mu} \cdot \vec{B}=- \: \gamma \, \vec{S} \cdot \vec{B} \; \; .$$ $$\hat{H}=- \: \gamma S_{z} B_{z} \; \; .$$ Although this Hamiltonian isn't very interesting because it is proportional to an operator we already know ( $\;S_{z}\;$ ) and because it depends on the magnitude of the external magnetic field, it is a very useful operator. The eigenvalues of the matrix for the Hamiltonian will be the energy values that you can measure. Note that …