Parity of function of multiple variables

In summary, the conversation revolved around the concept of parity in functions of multiple variables, specifically in the context of quantum mechanics. It was mentioned that parity is typically defined for functions of one variable, but there seemed to be a need to determine the parity of potentials in quantum mechanics that involve multiple variables. It was noted that for a potential of the form V(x,z) = xz, changing the sign of all variables is sufficient to make it an even function. The conversation also touched on the determination of which matrix elements of the form <n m l | V | n' m' l'> are zero for a spherically symmetric quantum mechanical system, using selection rules. It was mentioned that the potential V is an even function, and
  • #1
wuhtzu
9
0
Hi everyone

I was wondering how to determine the parity of a function of multiple variables.

Say the function is:

[tex]f(x,z) = xz[/tex]

How would I determine its parity?

For the above function it is true that #1 f(x,z) = f(-x,-z) but its also true that #2 [tex]\int_{-a}^{a}\int_{-a}^{a}xz \dx \dz = 0[/tex].

In one variable functions #1 would indicate an even function and #2 would indicate an odd function. So I guess you cannot directly extend the concept of odd and even function from one variable to multiple variables?

The ultimate need to answer such question is to determine the parity of potentials in quantum mechanics :)

Best regards
Wuhtzu
 
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  • #2
Parity is usually defined for functions of one variable.
 
  • #3
That is correct, but it doesn't remove the need to determine the parity of potentials suchs as

V(x) = E*x
V(x,z) = c*x*z
V(z) = B*L

in quantum mechanics :( Most of them is easy because they are only dependent on one direction ie. one variable, but some, like v(x,z) = c*x*z, is dependent on more...
 
  • #4
Parity (math term) is well defined for functions of one variable. I have never seen a definition for functions of more than one variable, other than for each variable separately.
 
  • #5
I will post what ever answer we get when our quantum mechanics professor gets back from vacation.

Thank you for looking
 
  • #6
So for this problem it turned out to be sufficient to just change sign of all variables, making it an even function.


The original problem if anyone is interested: Determine which matrix elements of the form

<n m l | V | n' m' l'> , n = n' = 2, m-m' = ? and l-l' = ?

was zero for a spherically symmetric quantum mechanical system by using selection rules.

The potential V was of the from V = x*z. The eigenstates of the spherically symmetric system has alternating even and odd parity with increasing l quantum number and the potential V = x*z is a rank k = 2 tensor.

This imposes the restriction [tex]l-l' = \Delta \le 2 \le 2[/tex].

Getting restrictions on m-m' is another story.

Thanks again.

Furthermore for [tex]<n m l | V | n' m' l'> \ne 0[/tex] the product of the wavefunctions <n m l | and |n' l' m'>, and the potential V has to be an even function - if it is odd it integrates to zero. Since we know that V is even both of the two wavefunctions (<n m l | or |n' m' l'>) have to have the same parity. Imposes futher restrictions on l-l' : [tex]\Delat l=0 , 2[/tex] since placing the l states 1 apart will cause one of them to be even and one of them to be odd (remember the alternating parity). But choosing l-'l = 0 or l-l'=2 causes both to be of the same parity which makes the total product even and hence the intergral non-zero.
 

1. What is meant by "parity of function of multiple variables"?

The parity of function of multiple variables refers to the behavior of a mathematical function when the input variables are changed. Specifically, it measures whether the function remains the same or changes when the input variables are interchanged or replaced with their negative values.

2. Why is the concept of parity important in mathematics?

The concept of parity is important in mathematics because it helps to identify symmetries and patterns in functions and equations. It also plays a crucial role in solving equations and proving theorems.

3. How is the parity of a function determined?

The parity of a function can be determined by analyzing the behavior of the function when the input variables are replaced with their negative values. If the function remains unchanged, it is said to have even parity. If the function changes sign, it is said to have odd parity.

4. What is the difference between even and odd parity?

The main difference between even and odd parity is that even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. This means that the graph of an even function remains unchanged when reflected across the y-axis, while the graph of an odd function changes sign when reflected across the y-axis.

5. How is the parity of a function used in real-world applications?

The concept of parity is used in various real-world applications, such as signal processing, cryptography, and error detection. In signal processing, even and odd functions are used to analyze signals and extract useful information. In cryptography, parity bits are used to detect errors in transmitted data. Overall, the concept of parity has numerous practical applications in fields such as engineering, physics, and computer science.

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