Any wave function can be written as a linear combination of ______a)Eigen Vectorsb)Eigen Valuesc)Eigen Functionsd)OperatorsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

A wave function describes the state of a particle. It does not have a physical significance. Moreover, it can be written as a linear combination of Eigen functions, i.e., Ψ(x) = AF(x) + BG(x).

Answer

Eigen Functions

An eigenfunction is a function that, when operated on by a linear operator, yields a scalar multiple of itself. In quantum mechanics, eigenfunctions are used to describe the wave functions of particles.

Linear Combination

A linear combination is a mathematical operation that involves adding together multiple vectors or functions, each multiplied by a scalar coefficient. The result is a new vector or function that is a combination of the original ones.

Wave Function

In quantum mechanics, the wave function represents the state of a particle or a system of particles. It is a mathematical function that describes the probability distribution of the particle's position or other observable properties.

Explanation

The statement 'Any wave function can be written as a linear combination of eigenfunctions' is a fundamental concept in quantum mechanics. It is known as the eigenfunction expansion theorem.

The eigenfunction expansion theorem states that any well-behaved wave function can be expressed as a linear combination of eigenfunctions of a given operator. These eigenfunctions are the solutions to the corresponding eigenvalue equation.

When we apply an operator to an eigenfunction, the result is a scalar multiple of the original eigenfunction. This scalar multiple is known as the eigenvalue. In other words, the eigenvalue represents the value that the operator 'eigenfunctions' multiply their original function by.

By expressing a wave function as a linear combination of eigenfunctions, we can break it down into simpler components. Each eigenfunction represents a particular state or mode of the system, and the coefficients in the linear combination determine the contribution of each eigenfunction to the overall wave function.

This concept is widely used in quantum mechanics to solve problems and analyze the behavior of particles and systems. By decomposing a complex wave function into eigenfunctions, we can study the properties and interactions of the system in a more manageable way.

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