Schrödinger's Equation: The Heart of Quantum Mechanics

Image description: The Schrödinger Equation allows us to understand the probabilistic behavior of subatomic particles and to calculate the possible states of a quantum system.

The Schrödinger Equation

The Schrödinger Equation, formulated in 1925 by the Austrian physicist Erwin Schrödinger (1887-1961), is a fundamental pillar of quantum mechanics. It provides a mathematical representation of the temporal evolution of quantum systems, linking the wave function to the system's energy. Unlike the laws of classical mechanics, which describe deterministic trajectories, the Schrödinger Equation allows us to understand the probabilistic behavior of subatomic particles.

Form and Significance of the Equation

The Schrödinger Equation is generally written in two forms: the time-dependent form and the time-independent form. In the general time-dependent form, the first term describes how the particle's wave function varies in space, related to its kinetic energy. The second term describes how the environment or external forces influence the particle. The Schrödinger Equation relates these two terms to determine the particle's total energy.

$$i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t)$$

• $\Psi(\mathbf{r}, t)$ is the wave function that describes the quantum state of the system at a given moment.
• $\hbar$ is the reduced Planck constant.
• $\hat{H}$ is the Hamiltonian operator that represents the total energy of the system (kinetic + potential).
• $i$ is an imaginary number which, when multiplied by itself, gives -1 (i²=-1). This allows the generalization of the real number system to include solutions to equations that cannot be solved within the set of real numbers, such as x²+1=0.

The wave function, $\Psi(\mathbf{r}, t)$, contains all the information about the quantum system, and its square gives the probability density of the particle's presence at a given position. Therefore, the equation establishes a relationship between energy, the particle's state, and its probability of location in space-time.

Physical Interpretation

The probabilistic interpretation of quantum mechanics, introduced by Max Born (1882-1970), is based on the idea that the wave function $\Psi$ does not give a precise position of a particle, but a probability of its presence in a given volume. Thus, the Schrödinger Equation allows us to calculate the possible states of a quantum system, as well as the probabilities of its different configurations.

Simplified Comparison with a Classical Object

If we take the analogy of a ball thrown into the air:

• Kinetic energy (½mv²) describes its motion through the air, which is the energy due to the object's velocity.
• Potential energy (mgh) depends on its position (height in a gravitational field).
• The sum of the two energies gives the total energy (E=½mv² + mgh).

The Schrödinger Equation does the same thing, but for a quantum particle with wave-like properties.

• The left-hand term of the Schrödinger Equation expresses that the temporal evolution of the quantum state is directly governed by the total energy of the system.
• The right-hand term represents the total energy of the system as a mathematical operator applied to the wave function.
• In simple terms: the change over time of a particle's quantum state is determined by its total energy. This is a quantum reformulation of energy conservation where temporal evolution is dictated by the combination of kinetic and potential energies encapsulated in \( \hat{H} \), the Hamiltonian operator, which represents the total energy of the system.

N.B.: The Schrödinger Equation, which describes the evolution of a quantum system, is itself a reflection of the system's symmetries. The invariance of this equation with respect to certain transformations (such as translations or rotations) is directly related to conservation laws according to Noether's theorem, where each continuous symmetry of a physical action corresponds to a conservation law (energy, charge, or momentum).

Applications and Consequences

The Schrödinger Equation is fundamental for understanding quantum phenomena such as the quantization of energy levels, stationary states, and wave-particle duality. It has led to major advances in fields such as quantum chemistry, solid-state physics, and particle physics. It is used to model atomic, molecular, and even some macroscopic systems subject to quantum interactions.

Conclusion

The Schrödinger Equation marks a break with the classical view of the world, where objects follow well-defined trajectories. In the framework of quantum mechanics, it introduces a new understanding of reality, where particles no longer exist as distinct objects, but as probabilities of being in a given location. This revolutionary vision has not only reshaped our understanding of the infinitely small, but it has also paved the way for cutting-edge technologies, such as transistors and semiconductors, lasers, electron and tunneling microscopy, materials science, medical imaging, quantum computers, quantum sensors, etc.