Eigenvalues and eigenfunctions in the Schrödinger equation

The concept of eigenvalue and eigenfunction is important in the field of quantum mechanics, particularly in the context of the Schrödinger equation. Quantum mechanics, an essential pillar of modern physics, describes the physical properties of nature at the smallest scales of the energy levels of atoms and subatomic particles. The Schrödinger equation, proposed by Erwin Schrödinger in 1925, is one of the key results of quantum mechanics and is a partial differential equation that describes how the quantum state of a physical system changes over time.

The Schrödinger equation is foundational to understanding the behaviour of quantum systems and how particles such as electrons behave in atomic or molecular frameworks. Within this equation, the notions of eigenvalue and eigenfunction naturally emerge, providing insight into the stationary states of systems, and helping to understand the quantisation of physical properties such as energy.

Understanding the Schrödinger equation

To understand the concept of eigenvalues and eigenfunctions, we first need to understand the basics of the Schrödinger equation. There are two forms of this equation: the time-dependent and time-independent Schrödinger equation. The time-independent equation is particularly useful when dealing with systems in a steady state.

        iħ ∂/∂t Ψ(x, t) = Ĥ Ψ(x, t)
    

The above equation is the time-dependent Schrödinger equation, where i is the imaginary unit, ħ is the reduced Planck constant, Ψ(x, t) represents the wave function of the quantum system, and Ĥ is the Hamiltonian operator corresponding to the total energy of the system. The equation describes how the wave function of a quantum system evolves with time.

For stationary states, which do not change with time, the time-independent Schrödinger equation is more relevant:

        ĤΨ(x) = ∆Ψ(x)
    

The time-independent Schrödinger equation is an eigenvalue equation where Ĥ is the operator, Ψ(x) is the eigenfunction, and E is the eigenvalue. This equation is important for finding the stationary states of a quantum system and their corresponding energy levels.

Explanation of eigenvalues and eigenfunctions

In mathematics, the eigenvalue problem involves an operator that acts on a function resulting in a scalar multiple of that function. In the context of the Schrödinger equation, Ĥ is an operator that represents the energy of the system, and when it acts on the wave function Ψ, the result is that Ψ is scaled by a factor of E. This is where the terms eigenfunction and eigenvalue arise.

Eigenfunctions

Consider the operator Ĥ applied to the wave function Ψ:

        ĤΨ(x) = ∆Ψ(x)
    

Here, Ψ(x) is called the eigenfunction of the operator Ĥ. An eigenfunction is a non-zero function that returns the same function (up to scalar multiplication) when the operator is applied. Eigenfunctions represent the possible states of a quantum mechanical system.

For a better understanding, imagine a function that acts as a mirror. Here, the function is the image reflected by the mirror, which is unchanged except for being flipped or scaled. In quantum mechanics, the actual "shape" or form of the wave function Ψ(x) may vary, but it remains basically "the same" with respect to the operator Ĥ, except for being scaled by E.

Eigenvalue

The term E in the equation is known as the eigenvalue. An eigenvalue is a scalar associated with the eigenfunctions and is specific to a particular eigenfunction of the operator. In quantum mechanics, these eigenvalues represent observable quantities such as energy.

When an eigenvalue is associated with an energy operator (Hamiltonian), it represents the energy level of the corresponding stationary state ψ(x). For example, in an atom, the eigenvalues of the Hamiltonian correspond to quantized energy levels that an electron can occupy.

Visual example: Harmonic oscillator

Consider the quantum harmonic oscillator, a common problem that illustrates the concepts of eigenvalues and eigenfunctions. The potential energy function for the harmonic oscillator is:

        v(x) = 1/2 m ω² x²
    

where m is the mass of the particle and ω is the angular frequency. The Hamiltonian for the quantum harmonic oscillator can be written as:

        Ĥ = -ħ²/2m d²/dx² + 1/2 m ω² x²
    

The eigenfunctions (ψ_n(x)) and eigenvalues (E_n) for a quantum harmonic oscillator are given by:

        ψ_n(x) = (1/√(2^nn!)) (mω/πħ)^(1/4) e^(-mωx²/2ħ) H_n(√(mω/ħ)x)
    

        E_n = ħω(n + 1/2)
    

where n is a non-negative integer, and H_n are Hermite polynomials. Each eigenfunction ψ_n(x) corresponds to a quantized energy level E_n. The quantum harmonic oscillator demonstrates how eigenfunctions and eigenvalues are essential for describing the quantum states and energy levels of a system.

Example: Hydrogen atom

Another classic example is the hydrogen atom, the simplest atom, where a single electron orbits a proton. Solving the Schrödinger equation for the hydrogen atom can give eigenfunctions that describe the possible orbits of the electron and eigenvalues that represent the energy levels of those orbits:

        Ĥψ = eψ
    

where the potential energy V is due to the Coulomb attraction between the negatively charged electron and the positively charged proton:

        v(r) = -e²/4πε₀r
    

Solving the Schrödinger equation with this potential energy gives the quantized energy levels and the corresponding wave functions (ψ). These eigenfunctions describe the shape and orientation of the electron's orbital paths, while the eigenvalues represent the corresponding energy levels.

Visualization of eigenfunction behavior

For visualization, imagine pulling a string fixed at both ends. The stationary patterns that form on this string as it is pulled correspond to eigenfunctions. These natural patterns reflect the many ways a stationary wave can form, similar to the different states of an electron in a quantum system. The wavelengths (related to the wave number k) are related to the fixed length of the string, reflecting the quantized nature of energy levels in quantum mechanics.

In this illustration, each curve represents a possible wave pattern on a fixed string. Such standing waves correspond to eigenfunctions, each of which has its own fixed wavelength (or frequency). In quantum mechanics, each standing wave function aligns with a specific energy level, or eigenvalue, which outlines the quantized states found within the electron configuration.

Conclusion

Eigenvalues and eigenfunctions are indispensable for understanding the mathematical framework of quantum mechanics, especially in relation to the Schrödinger equation. They allow physicists to understand the nature of quantum states, the quantized energies associated with specific particles, and to resolve specific wave functions. By conceptualizing these abstract ideas with visual and tangible analogies, such as harmonic oscillators or hydrogen atoms, the basic principles can become more accessible, providing insight into the complex dance of particles on the quantum scale.

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