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Schrödinger Equation Calculator

Solve the time-dependent and time-independent Schrödinger equations for quantum systems

Parameters

kgⓘ
Jⓘ
Jⓘ
sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Kinetic Energy:
0.00;J0.00;J
Energy (eV):
1.00;eV1.00;eV
Momentum:
0.00;kg⋅m/s0.00;kg·m/s
De Broglie Wavelength:
0.00;m0.00;m
Wave Number:
5123167222.81;m−15123167222.81;m⁻¹
Group Velocity:
593096.96;m/s593096.96;m/s
Angular Frequency:
1519267448809510.50;rad/s1519267448809510.50;rad/s
Phase Factor:
1519.27;rad1519.27;rad

Examples

Example 1: Free Electron

An electron with 1 eV kinetic energy in free space.

  • Energy (eV): 1.001.00
  • Kinetic Energy: 0.000.00
  • Momentum: 0.000.00
  • De Broglie Wavelength: 0.000.00
  • Group Velocity: 188000.00188000.00

Example 2: Electron in Potential

An electron in a 2 eV potential well with 3 eV total energy.

  • Energy (eV): 3.003.00
  • Kinetic Energy: 0.000.00
  • Momentum: 0.000.00
  • De Broglie Wavelength: 0.000.00
  • Group Velocity: 266000.00266000.00

Example 3: Proton in Field

A proton in a 1 MeV electric field.

  • Energy (eV): 1000000.001000000.00
  • Kinetic Energy: 0.000.00
  • Momentum: 0.000.00
  • De Broglie Wavelength: 0.000.00
  • Group Velocity: 13900000.0013900000.00

Visualization

Schrödinger Equation

The Schrödinger equation is the fundamental equation of quantum mechanics, describing how quantum systems evolve over time. It was formulated by Erwin Schrödinger in 1926 and is essential for understanding the behavior of particles at the atomic and subatomic level.

The time-dependent Schrödinger equation is: iℏ∂ψ/∂t = Ĥψ, where ψ is the wavefunction, Ĥ is the Hamiltonian operator, ℏ is the reduced Planck constant, and i is the imaginary unit. This equation describes how the wavefunction evolves in time.

The time-independent Schrödinger equation is: Ĥψ = Eψ, where E is the energy eigenvalue. This equation is used to find stationary states and energy levels of quantum systems. The wavefunction ψ represents the quantum state of the system.

The wavefunction ψ(x,t) contains all the information about a quantum system. The square of its absolute value, |ψ(x,t)|², gives the probability density of finding the particle at position x at time t. This is known as the Born interpretation.

The Schrödinger equation has solutions that exhibit wave-like behavior, including interference, tunneling, and quantization of energy levels. These phenomena are fundamentally different from classical physics and demonstrate the wave-particle duality of matter.

Important Note: When the particle energy equals the potential energy (E = V), the kinetic energy becomes zero, leading to zero momentum and infinite de Broglie wavelength. This represents a stationary particle. For meaningful quantum calculations, ensure the particle has sufficient kinetic energy.

Key Concepts

  • Time-Dependent Schrödinger Equation: iℏ∂ψ/∂t = Ĥψ
  • Time-Independent Schrödinger Equation: Ĥψ = Eψ
  • Wavefunction: ψ(x,t) - contains quantum state information
  • Probability Density: |ψ(x,t)|² - probability of finding particle
  • Hamiltonian Operator: Ĥ = -ℏ²/2m ∇² + V(x)
  • Energy Eigenvalues: Discrete energy levels in bound states

Real-World Applications

  • Atomic Physics: Understanding electron orbitals and energy levels
  • Molecular Physics: Chemical bonding and molecular spectroscopy
  • Solid State Physics: Electronic band structure and conductivity
  • Quantum Computing: Qubit states and quantum algorithms
  • Particle Physics: Fundamental particle behavior and interactions

Explore Further

More quantum mechanics tools

Physics Equations

Time-Dependent Schrödinger Equation:
iℏ∂ψ∂t=H^ψi\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi
Time-Independent Schrödinger Equation:
H^ψ=Eψ\hat{H}\psi = E\psi
Hamiltonian Operator:
H^=−ℏ22m∇2+V(x)\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x)
Probability Density:
P(x,t)=∣ψ(x,t)∣2P(x,t) = |\psi(x,t)|^2
Wavefunction Evolution:
ψ(x,t)=ψ(x,0)e−iEt/ℏ\psi(x,t) = \psi(x,0)e^{-iEt/\hbar}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Energy in eV

First, we convert the energy from Joules to electron volts:

Equation:

EeV=EJeE_{eV} = \frac{E_J}{e}

Calculation:

EeV=1.60e−191.60e−19=1.00 eVE_{eV} = \frac{1.60e-19}{1.60e-19} = 1.00 \text{ eV}

Explanation:

This gives the energy in more familiar units for quantum systems.

2

Step 2: Calculate Kinetic Energy

We calculate the kinetic energy from the total energy and potential:

Equation:

K=E−VK = E - V

Calculation:

K=1.60e−19−0.00e+0=1.60e−19 JK = 1.60e-19 - 0.00e+0 = 1.60e-19 \text{ J}

Explanation:

The kinetic energy is the difference between total energy and potential energy.

3

Step 3: Calculate Momentum

Using the kinetic energy, we find the momentum:

Equation:

p=2mKp = \sqrt{2mK}

Calculation:

p=2(9.11e−31)(1.60e−19)=5.40e−25 kg\cdotpm/sp = \sqrt{2(9.11e-31)(1.60e-19)} = 5.40e-25 \text{ kg·m/s}

Explanation:

This gives us the momentum of the particle.

4

Step 4: Calculate De Broglie Wavelength

Using the momentum, we find the de Broglie wavelength:

Equation:

λ=ℏp\lambda = \frac{\hbar}{p}

Calculation:

λ=1.05e−345.40e−25=1.95e−10 m\lambda = \frac{1.05e-34}{5.40e-25} = 1.95e-10 \text{ m}

Explanation:

This is the wavelength associated with the particle's wave-like behavior.

5

Step 5: Calculate Wave Number

The wave number is related to the momentum:

Equation:

k=pℏk = \frac{p}{\hbar}

Calculation:

k=5.40e−251.05e−34=5.12e+9 m−1k = \frac{5.40e-25}{1.05e-34} = 5.12e+9 \text{ m}^{-1}

Explanation:

The wave number determines the spatial frequency of the wavefunction.

6

Step 6: Calculate Group Velocity

The group velocity is the velocity of the particle:

Equation:

vg=pmv_g = \frac{p}{m}

Calculation:

vg=5.40e−259.11e−31=5.93e+5 m/sv_g = \frac{5.40e-25}{9.11e-31} = 5.93e+5 \text{ m/s}

Explanation:

This is the classical velocity of the particle.

7

Step 7: Calculate Angular Frequency

The angular frequency determines the time evolution:

Equation:

ω=Eℏ\omega = \frac{E}{\hbar}

Calculation:

ω=1.60e−191.05e−34=1.52e+15 rad/s\omega = \frac{1.60e-19}{1.05e-34} = 1.52e+15 \text{ rad/s}

Explanation:

This determines how fast the wavefunction oscillates in time.

8

Step 8: Calculate Phase Factor

The wavefunction evolves with a phase factor:

Equation:

ϕ=Etℏ\phi = \frac{Et}{\hbar}

Calculation:

ϕ=1.60e−19×1.00e−121.05e−34=1.52e+3 rad\phi = \frac{1.60e-19 \times 1.00e-12}{1.05e-34} = 1.52e+3 \text{ rad}

Explanation:

This phase factor describes how the wavefunction oscillates in time.

Frequently Asked Questions (FAQ)

What is the Schrödinger equation?

The Schrödinger equation is the fundamental equation of quantum mechanics that describes how quantum systems evolve over time. It relates the time evolution of the wavefunction to the system's energy and potential.

What is the difference between time-dependent and time-independent Schrödinger equations?

The time-dependent equation describes how the wavefunction changes over time, while the time-independent equation finds stationary states with definite energy values. The time-independent equation is used to find energy levels and eigenstates.

What is the physical meaning of the wavefunction?

The wavefunction ψ(x,t) contains all the information about a quantum system. The square of its absolute value, |ψ(x,t)|², gives the probability density of finding the particle at position x at time t.

What is the Hamiltonian operator?

The Hamiltonian operator Ĥ represents the total energy of the system. It includes both kinetic energy (-ℏ²/2m ∇²) and potential energy (V(x)) terms. It acts on the wavefunction to give the energy eigenvalue.

Why is the Schrödinger equation important?

The Schrödinger equation is essential for understanding atomic and molecular structure, chemical bonding, electronic properties of materials, and many other quantum phenomena. It's the foundation of modern quantum physics.

Practice MCQs

  1. The time-dependent Schrödinger equation is:
  2. The probability density is given by:
  3. The Hamiltonian operator includes:
  4. Energy eigenvalues in bound states are:
  5. The wavefunction ψ(x,t) represents:

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