Odd/Even functions and integration of them

In summary, the conversation is about the concept of parity of functions in quantum mechanics and how it applies to calculating the expectation value of position. The main confusion arises from the fact that the cosine function, which is considered an even function, becomes an odd function when squared and multiplied by the odd function x. This is due to the general rules of multiplying parity numbers.
  • #1
FaraDazed
347
2
I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed.

Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic concepts such as the parity of functions. I mean, I get the basic premise that if ##f(-x)=f(x)## then the function is even and if ##f(-x)=-f(x)## then it is odd, yet I still seem to struggle and things that are in my lecurers notes confuse me.

Essentially we are given a (wave) function which is essentially a cosine function ## \sqrt{\frac{2}{a}} cos(\frac{3 \pi x}{a}) ## to be precise, and looking at a past paper that has outline solutions, one question asks the parity of the function, I answered even since its a cos function and that is what the answer was. But then, it asks us to find the expectation value of position which requires squaring the given function and integrating over all space, but the question only has 2 marks so only quick sentence is needed, and the answer given in the solutions that it is zero because the integral is of an ODD function.

That is the bit that confuses me, how has it become an odd function, just simply by squaring it? It is still symmetric about the y-axis is it not? And if you put -x into either of the now two cosine functions, surely they both spit out the same number if x was inserted?
 
Mathematics news on Phys.org
  • #2
When you calculate the expectation value of position in one dimension, the integrand will be something like ##x|\psi(x)|^2##. ##x## is an odd function, if furthermore ##|\psi(x)|^2## is even, then the product of them will be?
 
  • Like
Likes FaraDazed
  • #3
blue_leaf77 said:
When you calculate the expectation value of position in one dimension, the integrand will be something like ##x|\psi(x)|^2##. ##x## is an odd function, if furthermore ##|\psi(x)|^2## is even, then the product of them will be?

Ah yeah. Sorry. I forgot about that pesky x, and was focusing only on the wavefunction and wavefunction squared thinking that I was missing something there!

Plus as a mature student, the time I learn about parity of numbers in general and multiplying them etc was about 16 years ago (probs longer)! Forgot that there were general rules about that! Gosh I feel stupid now haha.
 

1. What is an odd function?

An odd function is a type of mathematical function that satisfies the condition f(-x) = -f(x) for all values of x. This means that when the input is multiplied by -1, the output is also multiplied by -1. Visually, an odd function is symmetric about the origin, with the graph having rotational symmetry of 180 degrees.

2. What is an even function?

An even function is a type of mathematical function that satisfies the condition f(-x) = f(x) for all values of x. This means that when the input is multiplied by -1, the output remains the same. Visually, an even function is symmetric about the y-axis, with the graph having reflectional symmetry.

3. How do you determine if a function is odd or even?

To determine if a function is odd or even, you can use the substitution method. Substitute -x for x in the original function and simplify. If the resulting function is equal to the negative of the original function, then the function is odd. If the resulting function is equal to the original function, then the function is even.

4. How are odd and even functions integrated?

Odd functions are integrated from -a to a, where a is the limit of integration. Since the area under the curve on both sides of the y-axis is equal, the integral will be 0. Even functions are integrated from -a to a, but the resulting integral will be twice the integral from 0 to a. This is because the area under the curve on one side of the y-axis is reflected on the other side, resulting in a symmetrical shape.

5. What is the significance of odd and even functions in science?

Odd and even functions have many applications in science, particularly in physics and engineering. They are used to model many natural phenomena, such as waves and oscillations. In quantum mechanics, the wave functions of particles are either odd or even, which has important implications for their behavior and properties. Additionally, the symmetry properties of odd and even functions are utilized in many mathematical techniques and methods, making them essential tools for scientists.

Similar threads

MHB Fourier Series involving Hyperbolic Functions
    • General Math
Replies
9
Views
2K
I Calculating the inverse of a function involving the error function
    • General Math
Replies
3
Views
193
I Naming convention for "Functional Variance"?
    • General Math
Replies
7
Views
820
I Odd/even functions and fractional indices
    • General Math
Replies
28
Views
3K
I Can the same argument be used for both radians and degrees in the sine function?
    • General Math
Replies
3
Views
733
I What are the applications of inverses of vector functions?
    • General Math
Replies
17
Views
2K
B Decomposition of a function into even and odd parts
    • General Math
Replies
28
Views
5K
I Alternating Harmonic Numbers are cool, spread the word!
    • General Math
Replies
4
Views
384
B Helping a 5-Year-Old Integrate a Function in Kindergarten
    • General Math
Replies
1
Views
631
I Finding a Rational Function with data (Pade approximation)
    • General Math
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top