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I wish I could speed teach Calculus, Linear Algebra and Differential Equations... so the Quantum Mechanics could be made plain. It is much simpler than I think people realize.
Anyone have ideas on how to do that?
Of course, I could be biased in that I learned classical mechanics first. Teaching that may actually be the more difficult task. This too requires Calculus, Linear Algebra and Differential Equations.
For those brave enough to attempt a crash course in the three subjects:
Calculus:
Definition of a Limit: Limit -- from Wolfram MathWorld
Definition of a derivative (requires understandning limits): Derivative -- from Wolfram MathWorld
Definition of an integral (requires understandning limits): Riemann Integral -- from Wolfram MathWorld Wolfram Mathematica Online Integrator
The fundamental Theorem of Calculus: Fundamental Theorems of Calculus -- from Wolfram MathWorld
The Second Fundamental Theorem of Calculus: Second Fundamental Theorem of Calculus -- from Wolfram MathWorld
Taylor and Maclaurin Series: Taylor Series -- from Wolfram MathWorld Maclaurin Series -- from Wolfram MathWorld
Linear Algebra:
Matrices: Matrix -- from Wolfram MathWorld
Vector Spaces:Vector Space -- from Wolfram MathWorld
Inner Product Spaces:Inner Product Space -- from Wolfram MathWorld
Orthonormal Bases: Orthonormal Basis -- from Wolfram MathWorld
Eigenvalues and Eigenvectors: Eigenvalue -- from Wolfram MathWorld Eigenvector -- from Wolfram MathWorld
Linear Transformations: Linear Transformation -- from Wolfram MathWorld
Differential Equations (very much cookbook in it's approach, just follow the recepies):
ordinary differential equations: Ordinary Differential Equation -- from Wolfram MathWorld
partial differential equations: Partial Differential Equation -- from Wolfram MathWorld
Fourier Series: Generalized Fourier Series -- from Wolfram MathWorld
Fourier and Laplace Transforms: Fourier Transform -- from Wolfram MathWorld Laplace Transform -- from Wolfram MathWorld
Euler Method (Numerical Solution method): Euler Forward Method -- from Wolfram MathWorld
Runge-Kutta Method (More sophisticated numerical solution method): Runge-Kutta Method -- from Wolfram MathWorld
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
Lol I think you put at least three semester's worth of knowledge into one post.
My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
http://www.revoltingvegetables.com
For QM, I'd add a dash of complex numbers to that list.
You can't wait for inspiration. You have to go after it with a club. - Jack London
I've never really looked into QM even though I apparantly have the math background. I guess I'll have to check it out when I get the time.
My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
http://www.revoltingvegetables.com
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My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
http://www.revoltingvegetables.com
To quote a math grad student "quantum mechanics is pretty easy, its all just a linear [vector?] space". I heard a QM prof say essentially the same thing.
1) infinite-dimenionsal linear vector space [ie Hilbert space]
2) solve a modified wave equation for the particular potential [V] under question. the solution is your wave function psi, or at least a part of it [add spin or other things as needed]. this will amount to PDE, ODE, BVP type problems dealing with the Schrodinger equation
3) the construct psi star psi serves as your probability density. As a mathematical note from probability theory we assert normalization, ie the integral of psi star psi over all relevant space =1. That way when we multiply two ormore things together we still get 1 x 1...=1.
4) to compute any quantity of physical interest we compute the expectation value, [which from probability theory?] is integral [psi star * relevant operator representing quantity of interest* psi] integrated over all of relevant space. note: the allowed energy levels are an exception to the above, they come out of the SE itself. but x, x^2, p, p^2, E, etc are computed as expectation values
5) there are mathematical nuances that are relevant and can be shown, but are kinda just details for this discussion.
Overall operating motto "1) identify the potential V, 2) solve the relevant schrodinger equation to find the wave function psi 3) compute all physical quantities of interest via <psi, A, psi> where A is the operator for the physical quantity of interest
I don't know if I made things easier or harder, but there you go.