Multiplying an even function by another even function yields an even function, while the product of two odd functions is also even. An odd function multiplied by an even function results in an odd function. These properties are commonly used in expectation-value calculations in quantum mechanics due to the symmetry of wave functions.
An even function multiplied by another even function will result in an even function. For example, if we take the function x² (which is even) and multiply it by another even function like e-x², the resulting function will also be even. Similarly, an odd function times another odd function will produce an even function, such as in the case of x sin(x). However, when an odd function is multiplied by an even function, the result is an odd function, as illustrated by the product x e-x².
This understanding is crucial when considering the symmetries of functions, especially in the context of expectation-value calculations in quantum mechanics, where the properties of symmetric wave functions are exploited for simplification. Furthermore, recognizing the symmetry of wave functions helps with analytical calculations, such as knowing that the integral over all space of an odd function is zero.