Time-independent Schrödinger equation

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the smallest scales, such as molecules, atoms, and subatomic particles. At the heart of quantum mechanics is the Schrödinger equation, a mathematical equation used to describe how the quantum state of a physical system changes over time.

The time-independent Schrödinger equation (TISE) is a specific form of the Schrödinger equation that applies to systems with a time-independent Hamiltonian. It is particularly useful in solving quantum systems where the potential energy does not change with time. Let us take a deeper look at TISE, its implications, and examples of how it applies to various quantum systems.

Understanding the time-independent Schrödinger equation

The Schrödinger equation is important in predicting the behavior of particles at the quantum level. For stationary systems (those that do not change over time), we focus on the time-independent form of this equation, which is given as:

Hψ(x) = Eψ(x)

In this equation:

• H: denotes the Hamiltonian operator, corresponding to the total energy of the system.
• ψ(x): the wave function of the system, which defines the probability amplitude of the different configurations.
• E: The energy eigenvalue associated with the wave function.

The Hamiltonian operator H typically consists of the kinetic and potential energy operators. In one dimension, it is expressed as:

H = -ℏ²/(2m) d²/dx² + V(x)

Where:

• ℏ: the reduced Planck constant.
• m: mass of the particle.
• V(x): potential energy as a function of position x.

Derivation of TISE from the time-dependent Schrödinger equation

Starting with the time-dependent Schrödinger equation (TDSE):

iℏ ∂ψ(x,t)/∂t = Hψ(x,t)

Assuming separation of variables, where the wave function can be expressed as:

ψ(x,t) = φ(x)T(t)

Plugging this into TDSE, we separate the variables so that one equation depends on x and the other on t:

iℏ T'(t)/T(t) = Hφ(x)/φ(x) = E

The left part represents the time-dependent part, and the right part the location-dependent part. By setting both to a constant E (the 'separation constant'), we arrive at two different equations. The spatial part is TISE:

Hφ(x) = Eφ(x)

Examples and applications of the time-independent Schrödinger equation

Particle in a box

Imagine a particle confined in a one-dimensional box between x = 0 and x = L, with infinite potential walls. The potential V(x) inside the box is zero and outside is infinite. Solving TISE gives:

-ℏ²/(2m) d²ψ(x)/dx² = Eψ(x)

By applying the limiting conditions (ψ(0) = ψ(L) = 0), we get the solution:

ψ_n(x) = sqrt(2/L) sin(nπx/L)

The corresponding energy level is quantized:

E_n = n²π²ℏ²/(2mL²)

This quantization results in discrete energy levels that the particle can occupy.

Quantum harmonic oscillator

The quantum harmonic oscillator is a model that describes particles (such as springs and molecules) subject to a restoring force proportional to the displacement. For this problem, TISE takes a known potential:

V(x) = (1/2) mω²x²

Solving the TISE for the harmonic oscillator gives the wave function:

ψ_n(x) = A_n H_n(√(mω/ℏ)) e^(-(mω/2ℏ)x²)

where H_n are the Hermite polynomials, and the energies are equally spaced:

E_n = (n + 1/2) ℏω

Such systems illustrate important quantum mechanical concepts such as zero-point energy because of their ever-existing ground state energy (n = 0 state).

Physical interpretation and significance

The wave function ψ(x) represents the probability amplitude, and its squared modulus |ψ(x)|² gives the probability density of detecting a particle at position x. The quantization of energy levels arises naturally from the boundary conditions applied to the wave function solutions of TISE.

Systems described by TISE are considered time-independent because the energy does not change with time. This provides important insights into bound state problems, where particles are confined within a potential, such as electrons in atoms or quantum dots.

Solving the time-independent Schrödinger equation

Solving TISE involves identifying the Hamiltonian operator and applying appropriate boundary conditions. Here are the steps to solve TISE:

1. Define the Hamiltonian: Identify the kinetic and potential energy components. For example, H = -ℏ²/(2m) d²/dx² + V(x).
2. Apply boundary conditions: The conditions arise from the physical setup (e.g., a particle in a box where ψ(0) = ψ(L) = 0).
3. Solve differential equations: Calculate wavefunctions that satisfy the differential equation and boundary conditions.
4. Quantize energy: Calculate the energy levels from the obtained boundary conditions and equations.

TISE solutions often involve well-known mathematical functions (e.g., sine and cosine for boxes, Hermite polynomials for oscillators).

Importance in quantum mechanics

In quantum mechanics the time-independent Schrödinger equation is essential for separating the energy eigenvalues and eigenstates of a quantum system with a stationary potential. This makes it easier to solve complex problems by focusing on configurations where the energy description of the system remains stationary over time.

TISE solutions provide important information, such as:

• Energy quantization provides the basis for understanding atoms, molecular structure and transitions.
• This forms the basis for atomic orbitals and electron configurations.
• This allows the exploration of quantum phenomena such as tunneling, resonance, and superposition.

In conclusion, mastering the time-independent Schrödinger equation gives physicists the tools to predict and visualize how quantum particles behave in confined environments, thereby contributing profoundly to our understanding of the quantum world.

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