Odd/even symmetric/antisymmetric

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In summary, odd and even symmetry refer to the behavior of a function when it is reflected over the y-axis. A function has odd symmetry if f(-x) = -f(x) and has even symmetry if f(-x) = f(x). An example of a function with odd symmetry is f(x) = x^3. A function cannot have both odd and even symmetry, but can have no symmetry at all. These concepts are useful in various mathematical and scientific applications, such as in Fourier analysis, signal processing, and understanding natural phenomena in physics.
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Are odd/even wavefunctions the same as symmetric/antisymmetric wavefunctions ?
I have seen symmetric/antisymmetric applied to 2 identical particles but is this the same as the odd/even wavefunctions for example applied to a single particle in an infinite well ?
Thanks
 
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In the context that they are usually used, no.
Odd / even typically refer to the spatial symmetry of the wavefunction, whereas the notion of symmetric and antisymmetric states applied to identical particle systems refer to the exchange symmetry of the particles.

Odd / Even: ##\psi(x) = \pm \psi(-x)##
Symmetric / Antisymmetric: ##\psi(x_1, x_2) = \pm \psi(x_2, x_1)##
 
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1. What is the difference between odd and even symmetry?

Odd and even symmetry refer to the behavior of a function when it is reflected over the y-axis. In odd symmetry, the function is unchanged after being reflected, while in even symmetry, the function is unchanged after being reflected and rotated 180 degrees.

2. How can you tell if a function has odd or even symmetry?

A function has odd symmetry if f(-x) = -f(x) and has even symmetry if f(-x) = f(x). This means that if the function remains unchanged after being reflected over the y-axis, it has odd symmetry, and if it remains unchanged after being reflected and rotated 180 degrees, it has even symmetry.

3. What is an example of a function with odd symmetry?

An example of a function with odd symmetry is f(x) = x^3. When this function is reflected over the y-axis, it remains unchanged. For example, f(2) = 8 and f(-2) = -8, so f(-2) = -f(2), indicating odd symmetry.

4. Can a function have both odd and even symmetry?

No, a function cannot have both odd and even symmetry. A function can only have one type of symmetry - either odd or even. However, there are functions that have no symmetry at all.

5. How are odd and even symmetry used in mathematics and science?

Odd and even symmetry are useful in various mathematical and scientific applications, such as in Fourier analysis and signal processing. In physics, odd symmetry is often associated with anti-symmetric properties, while even symmetry is associated with symmetric properties. These concepts are also important in understanding the behavior of natural phenomena, such as electromagnetic fields and quantum mechanical systems.

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