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1.2: Getting Started
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/01%3A_Scipy_Tutorial/1.02%3A_Getting_Started
The factor of $4\pi\epsilon_{0}$ in the denominator is annoying to keep around, so we will adopt "computational units". This means that we'll rescale the potential, positions and/or the charges so t...The factor of $4\pi\epsilon_{0}$ in the denominator is annoying to keep around, so we will adopt "computational units". This means that we'll rescale the potential, positions and/or the charges so that, in the new units of measurement, $4\pi\epsilon_{0}=1$ . On Windows, in the window that was opened up by the IDLE (Python GUI) program, click on the menu-bar item File → New File; then type Ctrl-s (or click on File → New File) and save the empty file as potentials.py.
11: Discrete Fourier Transforms
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/11%3A_Discrete_Fourier_Transforms
$\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$ \[\mathrm{IDFT}\Big\{F_0, F... $\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$ $\mathrm{IDFT}\Big\{F_0, F_1, \dots, F_{N-1}\Big\} = \Big\{f_0, f_1, \dots, f_{N-1}\Big\} \qquad\mathrm{where}\quad f_m = \frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i \frac{mn}{N}}\, F_n.$
1.3: Modularizing the Code
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/01%3A_Scipy_Tutorial/1.03%3A_Modularizing_the_Code
In programming terminology, our program is insufficiently "modular". Ideally, we want to isolate the part of the program that computes the potential from the part that specifies the numerical inputs t...In programming terminology, our program is insufficiently "modular". Ideally, we want to isolate the part of the program that computes the potential from the part that specifies the numerical inputs to the calculation, like the positions and charges. As explained by the comments, we intend to use these for the positions of the particles, the charges of the particles, and the positions at which to measure the potential, respectively.
12.2: General Description
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/12%3A_Markov_Chains/12.02%3A_General_Description
In physics, we are often interested in using Markov processes to model thermodynamic systems, such that a stationary distribution represents the distribution of thermodynamic micro-states under therma...In physics, we are often interested in using Markov processes to model thermodynamic systems, such that a stationary distribution represents the distribution of thermodynamic micro-states under thermal equilibrium. (We'll see an example in the next section.) Knowing the stationary distribution, we can figure out all the thermodynamic properties of the system, such as its average energy.
2: Scipy Tutorial (Part 2)
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/02%3A_Scipy_Tutorial_(Part_2)
This is part 2 of the Scientific Python tutorial.
8.4: Example- Particle-in-a-Box Problem
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/08%3A_Sparse_Matrices/8.04%3A_Example-_Particle-in-a-Box_Problem
Notice that we call eigsh using the sigma parameter, telling the eigensolver to find the eigenvalues closest in value to $-1.0$ : This will find the lowest energy eigenvalues because, in this case, a...Notice that we call eigsh using the sigma parameter, telling the eigensolver to find the eigenvalues closest in value to $-1.0$ : This will find the lowest energy eigenvalues because, in this case, all energy eigenvalues are positive. (We use $-1.0$ instead of 0.0, because the algorithm does not work well when sigma is exactly zero.) If there is a negative potential present, we would have to find a different estimate for the lower bound of the energy eigenvalues, and pass that to sigma.
4.7: Why Complex Numbers?
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Complex_Methods_for_the_Sciences_(Chong)/04%3A_Complex_Numbers/4.07%3A_Why_Complex_Numbers
One very important feature possessed by complex numbers and not real numbers is that the complex numbers are algebraically closed. There do exist number systems more complicated than the complex numbe...One very important feature possessed by complex numbers and not real numbers is that the complex numbers are algebraically closed. There do exist number systems more complicated than the complex numbers, which are formulated not by algebraic extension but by discarding one or more of the usual rules of algebra. One big reason that complex numbers have proven to be so important and useful is that it’s easy to formulate a version of calculus for them.
2.5: Appendix
https://phys.libretexts.org/Learning_Objects/Demos_Techniques_and_Experiments/Physics_Laboratory_(Chong)/2%3A_Writing_a_Good_Lab_Report/2.5%3A_Appendix
Figure $\PageIndex{1}$ : Here, the “Apparatus” and “Procedure” sections are lifted from the lab manual. The equipment list and assembly instructions are irrelevant; only the assembled apparatus ought...Figure $\PageIndex{1}$ : Here, the “Apparatus” and “Procedure” sections are lifted from the lab manual. The equipment list and assembly instructions are irrelevant; only the assembled apparatus ought to be described. Any text appearing in a figure is assumed to be relevant, and should thus be legible; if the text is irrelevant, omit it. Also, the horizontal axis should have a proper title (text description and symbol, not just a unit).
6.4: D- Numerical Tensor Products
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/06%3A_Appendices/6.04%3A_D-_Numerical_Tensor_Products
\[\begin{align} \hat{A} \otimes | b\rangle &= \sum_{\mu m'} |\mu\rangle \, [\texttt{kron}(A^T, b)^T]_{\mu m'} \, \langle m'| && \leftrightarrow \;\; \texttt{kron}(A^T, b)^T\\ \hat{A} \otimes \langle b... $\begin{align} \hat{A} \otimes | b\rangle &= \sum_{\mu m'} |\mu\rangle \, [\texttt{kron}(A^T, b)^T]_{\mu m'} \, \langle m'| && \leftrightarrow \;\; \texttt{kron}(A^T, b)^T\\ \hat{A} \otimes \langle b| &= \sum_{m\mu'} |m\rangle \, [\texttt{kron}(A, b^*)]_{m\mu'} \, \langle \mu'| && \leftrightarrow \;\; \texttt{kron}(A, b^*).\end{align}$
9.5: Monte Carlo Integration
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/09%3A_Numerical_Integration/9.05%3A_Monte_Carlo_Integration
It requires a much larger number of samples in order to reach a level of numerical accuracy comparable to the other numerical integration methods. (For 1D integrals, Monte Carlo integration typically ...It requires a much larger number of samples in order to reach a level of numerical accuracy comparable to the other numerical integration methods. (For 1D integrals, Monte Carlo integration typically requires millions of samples, whereas Simpson's rule only requires hundreds or thousands of discretization points.) However, Monte Carlo integration outperforms discretization-based integration schemes when the dimensionality of the integration becomes extremely large.
3.9: Exercises
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/03%3A_Quantum_Entanglement/3.09%3A_Exercises
$\langle\mu| \, \big(|\mu'\rangle + |\mu'' \rangle\big) = \langle \mu|\mu'\rangle + \langle \mu|\mu''\rangle$ In Section 3.1, we defined a tensor product space $\mathscr{H}_A\otimes\mathscr{H}_B$ ... $\langle\mu| \, \big(|\mu'\rangle + |\mu'' \rangle\big) = \langle \mu|\mu'\rangle + \langle \mu|\mu''\rangle$ In Section 3.1, we defined a tensor product space $\mathscr{H}_A\otimes\mathscr{H}_B$ as the space spanned by the basis vectors $\{|\mu\rangle\otimes|\nu\rangle\}$ , where the $|\mu\rangle$ ’s are basis vectors for $\mathscr{H}_A$ and the $|\nu\rangle$ ’s are basis vectors for $\mathscr{H}_B$ .

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