Glossary

9.2 Stationary states and energy eigenfunctions

2 min read•july 25, 2024

Stationary states and energy eigenfunctions are key concepts in quantum mechanics. They represent unchanging solutions to the , like a particle in a box, and correspond to specific energy levels, such as orbitals.

These states have constant and time-independent expectation values for observables. Understanding their properties is crucial for grasping quantum behavior and predicting measurement outcomes in various systems.

Stationary States and Energy Eigenfunctions

Stationary states and energy eigenfunctions

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  • Stationary states remain unchanged over time, represent time-independent solutions to Schrödinger equation (particle in a box)
  • Energy eigenfunctions solve time-independent Schrödinger equation, correspond to specific energy levels (hydrogen atom orbitals)
  • Relationship links stationary states and energy eigenfunctions through ψ(x,t)=ψ(x)e−iEt/ℏ\psi(x,t) = \psi(x)e^{-iEt/\hbar}
  • Properties include constant probability density and time-independent expectation values for observables (position, momentum)

Probability density in stationary states

  • Probability density calculated as ∣ψ(x,t)∣2=∣ψ(x)∣2|\psi(x,t)|^2 = |\psi(x)|^2, remains constant over time for stationary states
  • ensures total probability equals 1: ∫−∞∞∣ψ(x)∣2dx=1\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1
  • Expectation values for observables computed using probability density
    • Position: ⟨x⟩=∫−∞∞x∣ψ(x)∣2dx\langle x \rangle = \int_{-\infty}^{\infty} x|\psi(x)|^2 dx
    • Momentum: ⟨p⟩=∫−∞∞ψ∗(x)(−iℏddx)ψ(x)dx\langle p \rangle = \int_{-\infty}^{\infty} \psi^*(x)(-i\hbar\frac{d}{dx})\psi(x) dx
    • Energy: ⟨E⟩=∫−∞∞ψ∗(x)Hψ(x)dx=E\langle E \rangle = \int_{-\infty}^{\infty} \psi^*(x)H\psi(x) dx = E
  • Time-independence of expectation values characterizes stationary states

Properties of energy eigenfunctions

  • ensures eigenfunctions with different energies don't overlap: ∫−∞∞ψm∗(x)ψn(x)dx=δmn\int_{-\infty}^{\infty} \psi_m^*(x)\psi_n(x) dx = \delta_{mn}
  • allows expressing any well-behaved function as linear combination of eigenfunctions: f(x)=∑ncnψn(x)f(x) = \sum_n c_n\psi_n(x)
  • occur in bound states (atomic energy levels), in scattering states (free particles)
  • of Hamiltonian operator guarantees real energy eigenvalues, leads to orthogonality of eigenfunctions

Superposition principle for Schrödinger equation

  • allows combining solutions to form new valid solutions
  • General solution expressed as Ψ(x,t)=∑ncnψn(x)e−iEnt/ℏ\Psi(x,t) = \sum_n c_n\psi_n(x)e^{-iE_nt/\hbar}
  • Coefficients cnc_n determined by initial conditions: cn=∫−∞∞ψn∗(x)Ψ(x,0)dxc_n = \int_{-\infty}^{\infty} \psi_n^*(x)\Psi(x,0) dx
  • Time evolution occurs independently for each
  • Probability interpretation: ∣cn∣2|c_n|^2 gives probability of measuring energy EnE_n
  • Wave packets constructed by superposing energy eigenfunctions, localizing particles in space (Gaussian wave packet)

Key Terms to Review (20)

Bound State: A bound state refers to a quantum state of a particle or system where it is confined to a specific region in space due to the presence of a potential that is strong enough to hold it in place. In such states, the energy of the system is lower than the energy of free particles, leading to quantized energy levels. This concept is crucial in understanding stationary states and energy eigenfunctions, as bound states correspond to stable configurations where particles can exist without being able to escape to infinity.
Completeness: Completeness refers to a property of a mathematical space where every Cauchy sequence converges to a limit that is also within that space. This concept is fundamental in various areas of mathematics as it ensures that limits and solutions behave predictably within the context of the space, allowing for well-defined operations like integration and differentiation. Completeness plays a crucial role in ensuring that certain mathematical structures, like inner product spaces and Hilbert spaces, can be fully utilized in applications such as solving differential equations and analyzing quantum systems.
Continuous Spectra: Continuous spectra refer to a type of spectral distribution that consists of a range of wavelengths or energies without any distinct lines or gaps. In quantum mechanics, this concept is significant as it illustrates how certain systems can possess a continuum of energy states rather than discrete energy levels. Continuous spectra often arise in situations where particles can occupy a vast array of states, like free particles or systems under certain conditions, making them a crucial aspect in understanding stationary states and energy eigenfunctions.
Dirichlet boundary conditions: Dirichlet boundary conditions specify the value of a function at the boundaries of a domain. These conditions are crucial in mathematical physics as they help define the behavior of solutions to differential equations, especially when examining eigenvalue problems and quantum systems. They ensure that the solution matches a given function along the boundary, thereby influencing the nature of eigenfunction expansions and the characteristics of stationary states in quantum mechanics.
Discrete Spectra: Discrete spectra refer to the specific, quantized set of energy levels that a quantum mechanical system can occupy. In the context of stationary states, each energy eigenfunction corresponds to a particular energy eigenvalue, leading to a distinct, well-defined energy state for a particle or system. These discrete energy levels emerge from the solutions of the Schrödinger equation and are fundamental in understanding how systems behave at the quantum level.
Energy Eigenfunction: An energy eigenfunction is a specific type of wave function that corresponds to a definite energy value in quantum mechanics. These functions are solutions to the time-independent Schrödinger equation and describe the state of a quantum system where the energy is well-defined, often associated with stationary states that do not change in time. Energy eigenfunctions are crucial for understanding the behavior of quantum systems, as they provide insight into the possible energy levels and spatial distributions of particles.
Expectation Value: The expectation value is a fundamental concept in quantum mechanics that represents the average outcome of a measurement for a given observable in a quantum state. It connects to the mathematical framework through linear operators and is central to understanding how measurements affect the state of a system over time, as well as how different states relate to energy and spin.
Fourier Transform: The Fourier transform is a mathematical operation that transforms a function of time (or space) into a function of frequency, essentially breaking down signals into their constituent frequencies. This tool is crucial for analyzing physical systems, as it connects time-domain and frequency-domain representations, enabling various applications in physics, engineering, and applied mathematics.
Free particle solution: The free particle solution refers to the mathematical description of a quantum particle that is not subject to any forces or potentials, allowing it to move freely in space. This solution is crucial for understanding the behavior of particles in quantum mechanics, as it serves as a foundational example of how wave functions evolve over time, leading to key concepts like stationary states and energy eigenfunctions.
Hermiticity: Hermiticity refers to a property of operators in quantum mechanics where the operator is equal to its own Hermitian conjugate. This characteristic ensures that the eigenvalues of the operator are real, which is essential for observables in quantum mechanics, such as position and momentum. Hermitian operators guarantee that physical measurements yield real values, linking closely to the concept of stationary states and energy eigenfunctions.
Hydrogen atom: The hydrogen atom is the simplest and most abundant element in the universe, consisting of one proton and one electron. Its simplicity makes it a fundamental building block for understanding atomic structure and quantum mechanics, particularly in the context of stationary states and energy eigenfunctions.
Neumann boundary conditions: Neumann boundary conditions are a type of boundary condition used in differential equations, where the derivative of a function is specified on the boundary of the domain. These conditions are crucial in various physical situations, such as heat conduction and fluid flow, as they relate to the rate of change or flux across a boundary rather than the values themselves. In the context of mathematical physics, these conditions play a significant role in Sturm-Liouville theory and energy eigenfunctions, helping to define how systems behave at their limits.
Normalization Condition: The normalization condition is a mathematical requirement that ensures a wave function describes a valid quantum state by ensuring that the total probability of finding a particle within the entire space is equal to one. This concept is fundamental in quantum mechanics, linking it to Hilbert spaces, where wave functions reside, and the significance of energy eigenfunctions, which represent stationary states. Properly normalizing a wave function ensures it adheres to the probabilistic interpretation of quantum mechanics.
Orthogonality: Orthogonality refers to the concept where two vectors or functions are perpendicular to each other in the context of an inner product space. This property is crucial in various fields, as it allows for the decomposition of functions into simpler components, aids in solving differential equations, and facilitates the representation of quantum states in Hilbert spaces.
Probability density: Probability density is a function that describes the likelihood of finding a particle in a specific region of space at a given time. It provides insight into how the probability of locating a particle is distributed across different locations, and is crucial for understanding the behavior of quantum systems. In quantum mechanics, the square of the absolute value of the wave function, known as the wave function probability density, determines the probabilities associated with various outcomes of measurements.
Quantum harmonic oscillator: The quantum harmonic oscillator is a fundamental model in quantum mechanics that describes a particle subject to a restoring force proportional to its displacement from an equilibrium position, resembling the classical harmonic oscillator but incorporating the principles of quantum mechanics. This model is essential for understanding various physical systems, such as vibrational modes in molecules and the behavior of particles in a potential well. The solutions to the quantum harmonic oscillator yield discrete energy levels, which are vital for exploring concepts like stationary states and the application of ladder operators.
Schrödinger Equation: The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It plays a crucial role in understanding the behavior of particles at the quantum level, allowing us to connect wave functions to observable properties like energy and momentum.
Separation of Variables: Separation of variables is a mathematical method used to solve differential equations by expressing a function as the product of functions, each dependent on a single variable. This technique transforms complex problems into simpler ones by isolating variables, which is crucial in solving ordinary and partial differential equations, eigenvalue problems, and quantum mechanical systems.
Stationary state: A stationary state refers to a quantum state of a system where the probability distribution associated with the system's properties does not change over time. This concept is crucial in understanding how quantum systems behave, as stationary states correspond to energy eigenfunctions that represent stable configurations of a quantum system, making them essential in both classical and quantum mechanics contexts.
Superposition Principle: The superposition principle states that in a linear system, the net response at a given time or position is the sum of the individual responses from all influencing factors. This principle is central to understanding wave functions and states in quantum mechanics, where a particle can exist simultaneously in multiple states until measured, allowing for complex behaviors like interference and entanglement.
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