added parameter estimation tutorial for schrodinger equation#79
added parameter estimation tutorial for schrodinger equation#79a-b-h-a-y-s-h-i-n-d-e wants to merge 2 commits into
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| Parameter Learning of the Schrödinger Equation | |||
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Please add context on what Schrödinger's equation models (the physics and mathematics)
| time-dependent Schrödinger equation using gradient descent in Physika. | ||
| The Schrödinger equation is a partial differential equation that governs | ||
| the evolution of the wave function of a non-relativistic quantum-mechanical | ||
| system. |
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Please briefly describe what a "non-relativistic quantum-mechanical
system" is
| \frac{\partial \psi}{\partial t} = -\frac{i}{\hbar} \left( -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi \right) | ||
| \end{align*} | ||
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| This is the RHS of the Schrödinger equation — the rate of change of the |
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I think we can remove this
| .. code-block:: text | ||
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| Nx : ℕ = 1024 | ||
| x : ℝ[Nx] = linspace(-200, 200, Nx) |
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Lets make sure this (and other tutorials) runs properly with the new type system
| hbar : ℝ = 1.0 | ||
| mass : ℝ = 1.0 | ||
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| The timestep is chosen using the CFL stability condition for the |
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Please describe whar CFL is
| Nt : ℕ = 3271 | ||
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| The Initial Condition — Gaussian Wave Packet |
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Lets remove "The"
| return history | ||
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| .. note:: |
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Let's implement this with physika
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| x0 : ℝ = -50.0 # initial position | ||
| k0 : ℝ = 2.0 # wavenumber (controls momentum) | ||
| sigma : ℝ = 10.0 # width of the wave packet |
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Can we use greek letters?
| psi0 = (1 / sigma*sqrt(3.14))**0.5 * exp(1j * k0 * x) * exp(-((x - x0)**2) / (2 * sigma**2)) | ||
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| Discretizing the RHS |
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What does RHS refer to? I understand is d𝜓/dt from above, but please lets make it more explicit.
| Discretizing the RHS | ||
| --------------------- | ||
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| We discretize space into :math:`N_x` points with uniform spacing |
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Please add an explanation about discretization, why is relevant? What is the purpose of discretization?
| \end{align*} | ||
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| When the wave packet hits this barrier, part of it is reflected and part | ||
| tunnels through — a purely quantum mechanical phenomenon with no classical |
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What is the purpose of this text "— a purely quantum mechanical phenomenon with no classical
analogue"? If its not adding, lets remove it
| When the wave packet hits this barrier, part of it is reflected and part | ||
| tunnels through — a purely quantum mechanical phenomenon with no classical | ||
| analogue. The ratio of transmitted to reflected probability depends | ||
| sensitively on :math:`V_0`, which is why it is a learnable parameter. |
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I'm not sure that "The ratio of transmitted to reflected probability depends
sensitively on :math:V_0" implies "a learnable parameter." but the parameter we are trying to learn
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| def adam(bh: ℝ, g: ℝ, m: ℝ, v: ℝ, t: ℝ, lr: ℝ) : ℝ[4]: | ||
| beta1: ℝ = 0.9 | ||
| beta2: ℝ = 0.999V: ℝ[Nx] = make_potential(1.8) |
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Looks like V: ℝ[Nx] = make_potential(1.8) is a new line
| scalar. | ||
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| .. figure:: /_static/tutorial_files/pred_results_plot.gif |
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Very nice results!!
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