Are Sums and Differences of Eigenfunctions Also Eigenfunctions?

In summary, f1 + f2 cannot be an eigenfunction of L and f1-f2 is an eigenfunction with eigenvalue a2.
  • #1
eit32
21
0
a) Consider a linear operator L with 2 different eigenvalues a1 and a2, with their corresponding eigenfunction f1 and f2. Is f1 + f2 also an eigenfunction of L? If so, what eigenvalue of L does it correspond to? If not, why not?

b) Answer the same question as in part (a) but for the difference of the 2 functions;
f1-f2.
 
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  • #2
eit32 said:
a) Consider a linear operator L with 2 different eigenvalues a1 and a2, with their corresponding eigenfunction f1 and f2. Is f1 + f2 also an eigenfunction of L? If so, what eigenvalue of L does it correspond to? If not, why not?

b) Answer the same question as in part (a) but for the difference of the 2 functions;
f1-f2.

Let's see:

L(f1 + f2) = L(f1) + L(f2) (because L is linear)

= a1 f1 + a2 f2

If f1 + f2 is an eigenfunction then we must have:

L(f1 + f2) = b (f1 + f2)

for some eigenvalue b. This means that:

a1 f1 + a2 f2 = b f1 + b f2 ------>

(a1 - b) f1 + (a2 - b) f2 = 0

which means that f1 and f2 are proportional to each other. However, that is impossible because then the iegenvalues a1 and a2 have to bethe same. So, we arrive at a contradiction and f1 + f2 cannot be an eigenfunction of L.

Part b) minus f2 is also an eigenfunction with eigenvalue a2. So, the result of a) also applies in this case.
 

Related to Are Sums and Differences of Eigenfunctions Also Eigenfunctions?

What are Eigenfunctions and Eigenvalues?

Eigenfunctions and Eigenvalues are mathematical concepts used in linear algebra and differential equations. An eigenfunction is a function that, when multiplied by a constant value, remains unchanged. An eigenvalue is the constant value that causes this unchanged behavior.

What is the significance of Eigenfunctions and Eigenvalues?

Eigenfunctions and Eigenvalues are essential in understanding how systems behave under certain conditions. They are commonly used in physics, engineering, and other fields to model and analyze systems such as vibrations, heat transfer, and quantum mechanics.

How do you calculate Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors can be calculated through a process called diagonalization. This involves finding the characteristic equation of a matrix, solving for its roots, and then using those roots to find corresponding eigenvectors. Alternatively, there are also various algorithms and software programs available to compute eigenvalues and eigenvectors.

What is the relationship between Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors are closely related. An eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of itself, which is the corresponding eigenvalue. In other words, an eigenvector is a direction in which a linear transformation has only a scaling effect.

Why are Eigenfunctions and Eigenvalues important in quantum mechanics?

In quantum mechanics, eigenfunctions and eigenvalues are used to describe the state of a quantum system. The squared magnitude of an eigenfunction represents the probability of finding a particle in a particular state, while the corresponding eigenvalue represents the energy of that state. Eigenfunctions and eigenvalues are also used to solve the Schrödinger equation, which describes the time evolution of a quantum system.

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