Eigen Operator, Eigen Function, Eigen Value & Normalization of Wave Function

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In quantum mechanics, operators act on wave functions. Some special wave functions retain their original form after an operator acts on them and only get multiplied by a constant value. These concepts are known as eigen functions, eigen values, and eigen equations.

1. Eigen Operator, Eigen Function, Eigen Value and Eigen Equation

When an operator acts on a function and produces the same function multiplied by a numerical constant, then:

  • The function is called an Eigen Function.
  • The numerical constant is called an Eigen Value.
  • The complete relation is called an Eigen Equation.
  • The mathematical operator is called an Eigen Operator.

General Form of an Eigen Equation

\[ \hat{O}\psi = \lambda\psi \]

Where:

  • \(\hat{O}\) = Operator
  • \(\psi\) = Eigen Function
  • \(\lambda\) = Eigen Value


2. Schrödinger Wave Equation as an Eigen Equation

The time-independent Schrödinger equation is an example of an eigen equation:

\[ \hat{H}\psi = E\psi \]

Where:

  • \(\hat{H}\) = Hamiltonian Operator
  • \(\psi\) = Eigen Function (Wave Function)
  • \(E\) = Energy Eigen Value

3. Example of an Eigen Function

Consider the function:

\[ y=e^{2x} \]

Let the operator be differentiation with respect to \(x\):

\[ \frac{d}{dx} \]

Applying the operator:

\[ \frac{dy}{dx} = \frac{d}{dx}\left(e^{2x}\right) = 2e^{2x} \]

The original function \(e^{2x}\) remains unchanged except for multiplication by the constant value 2.

\[ \frac{d}{dx}\left(e^{2x}\right) = 2e^{2x} \]

Therefore:

  • \(\dfrac{d}{dx}\) is the Eigen Operator.
  • \(e^{2x}\) is the Eigen Function.
  • \(2\) is the Eigen Value.
  • The complete relation is an Eigen Equation.

4. Example That Is Not an Eigen Function

Consider:

\[ y=e^{-4x^{2}} \]

Applying the differentiation operator:

\[ \frac{dy}{dx} = \frac{d}{dx}\left(e^{-4x^{2}}\right) = -8x\,e^{-4x^{2}} \]

The result contains the variable \(x\) along with the original function. Since the multiplying factor is not a constant numerical value, this is not an eigen equation.

Reason:

\[ -8x \]

is not a constant eigen value because it depends on the variable \(x\).

5. Normalized Wave Function

In quantum mechanics, the probability of finding a particle in a small volume element is represented by:

\[ \psi\psi^{*}\,dV \]

or

\[ \psi\psi^{*}\,dx\,dy\,dz \]

where:

  • \(\psi\) = Wave Function (Eigen Function)
  • \(\psi^{*}\) = Complex Conjugate of the Wave Function

Since a particle must exist somewhere in the entire space, the total probability of finding the particle must be equal to 1 (or 100%).

6. Normalization Condition

A wave function is said to be normalized if the integral of the product of the wave function and its complex conjugate over the entire space is equal to unity.

\[ \int_{-\infty}^{+\infty} \psi\psi^{*}\,dx\,dy\,dz = 1 \]

This equation is called the Normalization Condition.

The limits from \(-\infty\) to \(+\infty\) represent the entire available space. The integral must be equal to unity because the particle must exist somewhere within that space.

Key Points to Remember

  • An eigen function remains unchanged in form after an operator acts on it.
  • The multiplying constant is called the eigen value.
  • The relation between operator, eigen function, and eigen value is called an eigen equation.
  • The Schrödinger equation is an important eigen equation.
  • \(\psi^{*}\) represents the complex conjugate of the wave function.
  • A normalized wave function satisfies the condition that the total probability of finding the particle is equal to 1.
Eigen Operator, Eigen Function, Eigen Value & Normalization of Wave Function


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