Origin of the Schrödinger equation

The development of quantum mechanics, culminating in the Schrödinger equation, was built on a series of groundbreaking discoveries in classical physics. To understand the origins of the Schrödinger equation, we need to start with the classical equations that describe particles and waves, and then follow the historical path that led to their unification in quantum mechanics.

Classical Equations for Particles and Waves

1. Classical Equation for Particles: Newton's Second Law and Hamiltonian Mechanics

• Newton’s Second Law:

  $$F=ma$$

  Where:

  • $F$ is the force acting on a particle.
  • $m$ is the mass of the particle.
  • $a$ is the acceleration of the particle.

• Hamiltonian Function: The Hamiltonian in classical mechanics represents the total energy of a system, which is the sum of kinetic and potential energy.

  $$H = T + V = \frac{p^2}{2m} + V(x$$

  Where:

  • $T$ is the kinetic energy $\left(\frac{p^2}{2m}\right)$.
  • $V(x)$is the potential energy as a function of position $x$.
  • $p=mv$is the momentum of the particle.

2. Classical Equation for Waves: The Wave Equation

• Wave Equation: $$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$$ Where:
  • $y(x,t)$ is the wave function, representing the displacement at position $x$ and time $t$.
  • $v$ is the velocity of the wave.

This equation describes how waves propagate through a medium.

From Classical Mechanics to Quantum Mechanics

1. Planck’s Quantization of Energy

The classical description of black body radiation failed to explain the observed spectrum, particularly at high frequencies. Max Planck resolved this by proposing that energy is quantized, introducing the concept of energy quanta:

Where:

• $E$ is the energy of a quantum.
• $h$ is Planck's constant.
• $\nu$ is the frequency of radiation.

This idea of quantized energy marked the first step toward quantum theory.

2. Einstein’s Photoelectric Effect and Wave-Particle Duality

Einstein extended Planck’s idea to explain the photoelectric effect, suggesting that light itself is made up of quanta (photons). Each photon has energy:

This revealed the dual nature of light—exhibiting both wave-like and particle-like properties—laying the groundwork for wave-particle duality.

3. De Broglie’s Hypothesis

Louis de Broglie proposed that not only light but all matter exhibits wave-particle duality. He suggested that particles like electrons have an associated wavelength:

Where:

• $\lambda$ is the wavelength associated with a particle.
• $p$ is the momentum of the particle.

This idea was pivotal in developing a wave equation that could describe particles.

Deriving the Schrödinger Equation

Using the classical Hamiltonian and de Broglie’s hypothesis, Schrödinger formulated his famous equation:

1. Wave Function and the Hamiltonian Operator

In quantum mechanics, the wave function $\psi (x,t)$ replaces the classical notion of a definite trajectory. The Hamiltonian $H$ becomes an operator acting on $\psi$.

The quantum mechanical operators for momentum $p$ and energy $E$ are:

Where $\mathrm{\hslash }$ is the reduced Planck’s constant.

2. Substituting into the Hamiltonian

Substitute the momentum operator into the classical Hamiltonian:

Expanding this gives:

3. Formulating the Schrödinger Equation

The time-dependent Schrödinger equation is obtained by applying the Hamiltonian operator to the wave function $\psi (x,t)$:

This equation describes how the quantum state of a system evolves over time.

Conclusion

The Schrödinger equation is the culmination of a series of developments in classical and quantum physics. It unifies the wave and particle descriptions, originally separate in classical mechanics, into a single framework that accurately describes the behavior of quantum systems. From Planck’s quantization to Einstein’s photoelectric effect, de Broglie’s wave-particle duality, and finally, Schrödinger’s formulation, this journey reflects the evolution of our understanding of the fundamental nature of reality.