In quantum mechanics, mathematical operators are used to represent measurable physical quantities such as energy, momentum, and position. When an operator acts on a particular wave function, the result may have exactly the same functional form as the original wave function, differing only by a constant multiplier.
Such a wave function is called an eigen function, and the corresponding constant is called an eigen value. The mathematical relationship connecting the operator, eigen function, and eigen value is known as an eigen equation.
1. Eigen Operator, Eigen Function, Eigen Value and Eigen Equation
Consider an operator represented by Ô acting on a function ψ. If the operation gives the original function multiplied by a constant, the relationship can be written as:
Ôψ = λψ
In this equation:
- Ô represents the operator.
- ψ represents the eigen function.
- λ represents the eigen value.
The important feature is that the function ψ retains its original mathematical form after the operator acts on it. Only its magnitude or amplitude is changed by the constant factor λ.
Definitions at a Glance
| Term | Meaning |
|---|---|
| Operator | A mathematical operation used to represent a physical observable. |
| Eigen Function | A function that retains its form when the operator acts on it, apart from multiplication by a constant. |
| Eigen Value | The constant by which the eigen function is multiplied. |
| Eigen Equation | The equation that relates the operator, eigen function, and eigen value. |
2. Schrödinger Wave Equation as an Eigen Equation
One of the most important eigen equations in quantum mechanics is the time-independent Schrödinger equation. It can be written as:
Ĥψ = Eψ
Here:
- Ĥ is the Hamiltonian operator.
- ψ is an eigen function of the Hamiltonian operator.
- E is the corresponding energy eigen value.
The Hamiltonian operator represents the total energy of the quantum system. Therefore, when a wave function is an eigen function of the Hamiltonian operator, the corresponding eigen value represents an allowed energy of the system.
This is why eigen values are particularly important in quantum mechanics: they connect the mathematical description of a quantum system with measurable physical quantities.
3. Example of an Eigen Function
A simple mathematical example can be obtained using the differentiation operator.
Consider the function:
y = e2x
Let the operator be differentiation with respect to x:
d/dx
Applying the operator to the function gives:
d/dx (e2x) = 2e2x
Notice that the resulting function is still e2x. It has only been multiplied by the constant 2.
Therefore:
- d/dx = operator
- e2x = eigen function
- 2 = eigen value
- (d/dx)e2x = 2e2x = eigen equation
This example demonstrates the defining characteristic of an eigen function: the operation changes the function only by a constant multiplicative factor.
Wave Function and Probability in Quantum Mechanics
In quantum mechanics, a wave function ψ contains information about the state of a particle. However, the wave function itself is not directly interpreted as a probability.
The probability density is related to the product of the wave function and its complex conjugate:
ψ*ψ = |ψ|²
For a particle described in three-dimensional space, the probability of finding the particle within a small volume element dV is:
|ψ|² dV
Since the wave function may be complex, ψ* denotes its complex conjugate. The product ψ*ψ is real and non-negative, which makes it suitable for representing probability density.
Normalization of a Wave Function
A physically acceptable wave function must be normalized so that the total probability of finding the particle somewhere in the allowed space is equal to one.
For a three-dimensional system, the normalization condition is:
∫ ψ*ψ dV = 1
Since:
dV = dx dy dz
the condition can also be written as:
∫∫∫ ψ*ψ dx dy dz = 1
For a one-dimensional system, the corresponding condition is:
∫−∞+∞ ψ*ψ dx = 1
The limits from −∞ to +∞ indicate that the integration covers the complete one-dimensional space.
What Does Normalization Mean?
Normalization simply means adjusting the wave function so that the total probability of finding the particle somewhere in the entire allowed region is exactly 1, or 100%.
If a wave function is not initially normalized, it can often be multiplied by an appropriate normalization constant so that it satisfies the normalization condition.
Important Difference: Eigen Equation vs Normalization Condition
These two ideas are often studied together in quantum mechanics, but they describe different mathematical requirements.
| Concept | Mathematical Form | Purpose |
|---|---|---|
| Eigen Equation | Ôψ = λψ | Identifies an eigen function and its corresponding eigen value. |
| Schrödinger Eigen Equation | Ĥψ = Eψ | Relates an energy eigen function to its allowed energy eigen value. |
| Normalization Condition | ∫ψ*ψ dV = 1 | Ensures that the total probability of finding the particle is unity. |
Quick Exam Revision
| Term | Remember This |
|---|---|
| Eigen Operator | The operator involved in an eigen equation. |
| Eigen Function | A function whose form remains unchanged apart from multiplication by a constant. |
| Eigen Value | The constant multiplier obtained when the operator acts on the eigen function. |
| Eigen Equation | Ôψ = λψ |
| Hamiltonian Eigen Equation | Ĥψ = Eψ |
| Probability Density | |ψ|² = ψ*ψ |
| Normalization | ∫ψ*ψ dV = 1 |
Common Exam Mistakes
- Do not confuse an eigen value with an eigen function. The eigen value is the constant multiplier, while the eigen function is the function being operated on.
- A multiplying factor is an eigen value only when it is a constant independent of the variables of the function.
- Do not write ψ alone as the probability density. The probability density is given by ψ*ψ or |ψ|².
- The normalization condition requires the total probability over the complete allowed space to be equal to 1.
- In the Schrödinger eigen equation, the energy E is the eigen value associated with the Hamiltonian operator.
Conclusion
Eigen functions and eigen values are fundamental concepts in quantum mechanics because they connect mathematical operators with measurable physical quantities. When an operator acts on an eigen function, the function retains its form and is multiplied only by its corresponding eigen value.
The time-independent Schrödinger equation, Ĥψ = Eψ, is a central example. Here, the Hamiltonian operator acts on the wave function and produces the same wave function multiplied by the energy eigen value. Along with this concept, normalization ensures that the total probability of locating the particle throughout the allowed space is equal to one.
Understanding these relationships makes later topics such as quantum numbers, allowed energy levels, expectation values, and measurement in quantum mechanics much easier to follow.
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.
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