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Boltzman Oscillation
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Homework Statement
Given:
Ψ and Φ are orthonormal find
(Ψ + Φ)^2
Homework Equations
None
The Attempt at a Solution
Since they are orthonormal functions then can i do this?
(Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
Yes. Now go ahead. Multiply it and apply what you know.Boltzman Oscillation said:Homework Statement
Given:
Ψ and Φ are orthonormal find
(Ψ + Φ)^2
Homework Equations
None
The Attempt at a Solution
Since they are orthonormal functions then can i do this?
(Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
fresh_42 said:Yes. Now go ahead. Multiply it and apply what you know.
Yes. I was asked to normalize and equation and it contained two orthonormal functions. I forgot how important the absolute value is.Delta2 said:There are some typos in this thread, most important one is that we want to find
##|\Psi+\Phi|^2## not just ##(\Psi+\Phi)^2## , right?
Technically, it is not the absolute value which counts! It is only a common indication, that ##\psi \chi = \langle \psi , \chi \rangle## is meant to be the multiplication. One could as well simply define the multiplication given by the inner product and then no absolute value would be necessary. So it's the way the square has to be interpreted which counts, not the shape of the brackets.Boltzman Oscillation said:I forgot how important the absolute value is.
I thought the absolute value's function was to make every section of the function positive so that the integral over that function doesn't cancel out. Kinda like taking the energy of a wave. Right?fresh_42 said:Technically, it is not the absolute value which counts! It is only a common indication, that ##\psi \chi = \langle \psi , \chi \rangle## is meant to be the multiplication. One could as well simply define the multiplication given by the inner product and then no absolute value would be necessary. So it's the way the square has to be interpreted which counts, not the shape of the brackets.
No, this is not right. It only makes the resulting values positive, not the individual function values. The square does the same.Boltzman Oscillation said:I thought the absolute value's function was to make every section of the function positive so that the integral over that function doesn't cancel out. Kinda like taking the energy of a wave. Right?
The square of the sum of two orthonormal functions refers to the mathematical operation of multiplying the sum of two orthonormal functions by itself. In other words, it is the product of the sum of two orthonormal functions with itself.
To calculate the square of the sum of two orthonormal functions, you first need to add the two functions together. Then, you multiply the resulting sum by itself. This can be represented by the equation (f+g)^2 = (f+g)(f+g).
Orthonormal functions are important in this context because they have a special property where their inner product is equal to 0, except when the two functions are the same. This allows for easier calculations and simplification of equations.
Some common examples of orthonormal functions include the sine and cosine functions, the Legendre polynomials, and the Hermite polynomials. Other examples include the Fourier basis functions and the Walsh basis functions.
The square of the sum of two orthonormal functions has various applications in mathematics and physics. It is commonly used in Fourier analysis, signal processing, and quantum mechanics. It is also used in solving differential equations and in the study of orthogonal polynomials.