Wavefunctions
- group Representations of the Infinite Square Well
group Small Group Activity
120 min.
Representations of the Infinite Square Well
Quantum Fundamentals 2023 (3 years) - group Spin-1 Time Evolution
group Small Group Activity
120 min.
Spin-1 Time Evolution
Quantum Fundamentals 2023 Students do calculations for time evolution for spin-1. - group Fourier Transform of a Gaussian
- group Fourier Transform of a Derivative
- group Fourier Transform of a Shifted Function
- assignment Fourier Transform of Cosine and Sine
assignment Homework
Fourier Transform of Cosine and Sine
Periodic Systems 2022- Find the Fourier transforms of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
- Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
- In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
- assignment Approximating a Delta Function with Isoceles Triangles
assignment Homework
Approximating a Delta Function with Isoceles Triangles
Static Fields 2023 (6 years)Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]
Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].
- Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
- Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).
- assignment One-dimensional gas
assignment Homework
One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of \(N\) particles, each of mass \(M\), confined to a one-dimensional line of length \(L\). The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature \(T\). You may assume that the temperature is high enough that \(k_B T\) is much greater than the ground state energy of one particle. - group Fourier Transform of a Plane Wave
- keyboard Sinusoidal basis set
keyboard Computational Activity
120 min.
Sinusoidal basis set
Computational Physics Lab II 2023 (2 years)inner product wave function quantum mechanics particle in a box
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions. -
Quantum Fundamentals 2023 (3 years)
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
- normalize the wave function,
- plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
- find the probability that the particle is measured to be in the range \(0<x<1\).
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Media & Figures
