The Schrödinger Equation

At the core of classical physics lies Newton's second law of motion, giving us the tools to predict the trajectory of a thrown baseball or the orbit of a planet. In the quantum realm, however, classical rules break down. Particles do not have simple, deterministic trajectories. To understand the universe at its most fundamental level, we must turn to the quantum equivalent of Newton's laws: the Schrödinger Equation.

1.1 Expressing the Equation

The Schrödinger equation comes in two primary forms, depending on whether the physical environment of the quantum system changes over time.

The Time-Dependent Schrödinger Equation (TDSE)

This is the most general form, describing how a quantum system's state evolves as time passes:

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

• $i$: The imaginary unit ($\sqrt{-1}$). Its fundamental presence means quantum mechanics inherently relies on complex mathematics.
• $\hslash$: The reduced Planck constant, which dictates the incredibly small scale at which quantum effects become noticeable.
• $\mathrm{\Psi }(\mathbf{r},t)$: The Wave Function. This is the mathematical description of the quantum state of the system at position $\mathbf{r}$ and time $t$. While it lacks direct physical substance, its absolute square ($|\mathrm{\Psi }{|}^2$) yields the probability of finding a particle in a specific location.
• $\hat{H}$: The Hamiltonian operator. This mathematical instruction represents the total energy of the system, combining both kinetic and potential energy.

The Time-Independent Schrödinger Equation (TISE)

If the forces or potential acting on the particle are stable and do not change with time, the equation simplifies. This form is used to find the stable, allowed energy levels (or "stationary states") of a system:

$$\hat{H}\psi (\mathbf{r})=E\psi (\mathbf{r})$$

• $E$: The specific, quantized energy levels the system is allowed to occupy.

1.2 Origins: A Logical Leap

The Schrödinger equation was not strictly "derived" from older mathematical proofs. Instead, it was postulated by Austrian physicist Erwin Schrödinger in 1926. Much like Newton's laws, it is accepted as a fundamental truth of nature because its predictions flawlessly match experimental reality.

However, Schrödinger built this equation using a logical trail of theoretical breakthroughs.

Step A: Wave-Particle Duality

In 1924, Louis de Broglie proposed that if light waves can act as discrete particles (photons), then physical matter (like electrons) might act as waves. He linked a particle's physical momentum ($p$) to a specific wavelength ($\lambda$) using Planck's constant ($h$):

$$\lambda =\frac{h}{p}$$

Step B: The Need for a "Wave Equation"

If electrons behave like waves, physics required a new mathematical equation to describe how these "matter waves" propagate and interact, just as classical equations describe sound or water waves. In wave physics, the simplest wave traveling in one direction is the plane wave, written in complex exponential form as:

$$\mathrm{\Psi }(x,t)=A{e}^{i(kx-\omega t)}$$

where $A$ is the amplitude, $k$ is the wave number (spatial frequency), and $\omega$ is the angular frequency (temporal frequency). This expression will serve as the starting point for translating classical quantities into quantum operators.

Step C: Classical Energy Conservation

Schrödinger began with the classical physics formula for the total energy ($E$) of a particle, which is the sum of its kinetic energy and potential energy ($V$):

$$E=\frac{p^2}{2m}+V$$

where $p$ is momentum and $m$ is mass.

Step D: The Quantum Bridge

To translate this classical equation into the quantum realm, Schrödinger applied mathematical "operators" to it, allowing the equation to act on de Broglie's matter waves. In quantum mechanics, momentum and energy are not static numbers; they are translated into instructions (derivatives) that tell the wave function what to do.

To understand why this is the case, consider a "free particle" moving through empty space described by the plane wave from Step B. Recall that this wave is characterized by two values:

• $k$ (Wave Number): Defined as $k=2\pi /\lambda$ (measured in radians per meter). It describes how many wave cycles fit into a unit of space.
• $\omega$ (Angular Frequency): Defined as $\omega =2\pi f=2\pi /T$ (measured in radians per second). It describes how fast the wave oscillates in time.

The Role of $2\pi$

Why the $2\pi$? Mathematically, a full cycle of a sine wave or a complex exponential ($e^{i\theta }$) repeats every $2\pi$ radians. By multiplying the spatial frequency ($1/\lambda$) and temporal frequency ($f$) by $2\pi$, we "normalize" the physical units of the wave so they fit perfectly into the geometry of a circle — which is precisely why the plane wave uses the complex exponential form $\mathrm{\Psi }(x,t)=A{e}^{i(kx-\omega t)}$.

Substituting Quantum Values

We can now bridge de Broglie's and Einstein's discoveries into this wave. We use the reduced Planck constant ($\hslash =h/(2\pi )$), which conveniently absorbs that $2\pi$ factor:

1. For Momentum ($p$):

   $$p=\frac{h}{\lambda }⟹p=\frac{h}{2\pi }\cdot \frac{2\pi }{\lambda }⟹p=\hslash k⟹\mathbf{k}\mathbf{=}\frac{\mathbf{p}}{\hslash }$$

2. For Energy ($E$):

   $$E=hf⟹E=\frac{h}{2\pi }\cdot 2\pi f⟹E=\hslash \omega ⟹\omega \mathbf{=}\frac{\mathbf{E}}{\hslash }$$

By substituting these expressions for $k$ and $\omega$ back into our plane wave equation, we get the wave function in terms of the particle's physical properties:

$$\mathrm{\Psi }(x,t)=A\exp \left[i(\frac{p}{\hslash }x-\frac{E}{\hslash }t)\right]=A\exp [\frac{i}{\hslash }(px-Et)]$$

Deriving the Energy Operator

If we want to extract the Energy ($E$) from that wave function, we take the partial derivative with respect to time ($t$). Using the chain rule, this brings down a factor from the exponent:

$$\frac{\partial }{\partial t}\mathrm{\Psi }=(-\frac{iE}{\hslash })\mathrm{\Psi }$$

To isolate $E$, we multiply both sides by $i\hslash$:

$$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=E\mathrm{\Psi }$$

Therefore, the quantum operator for energy is defined as: $E\to i\hslash \partial /\partial t$

Deriving the Momentum Operator

Similarly, to extract momentum ($p$), we take the derivative with respect to position ($x$):

$$\frac{\partial }{\partial x}\mathrm{\Psi }=\left(\frac{ip}{\hslash }\right)\mathrm{\Psi }$$

Multiplying both sides by $-i\hslash$ isolates $p$:

$$-i\hslash \frac{\partial }{\partial x}\mathrm{\Psi }=p\mathrm{\Psi }$$

Therefore, the quantum operator for momentum in one dimension is $-i\hslash \partial /\partial x$, which extends to three dimensions using the gradient operator: $p\to -i\hslash \mathrm{\nabla }$.

1.3 The General Formula and Its Limits

By substituting these quantum operators back into the classical energy conservation equation ($E=p^2/(2m)+V$) and applying them to the wave function $\mathrm{\Psi }$, we arrive at the full, unpacked Time-Dependent Schrödinger Equation:

$$i\hslash \frac{\partial \mathrm{\Psi }}{\partial t}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }+V\mathrm{\Psi }$$

This is the most general formulation for a single, non-relativistic particle of mass $m$ moving through 3D space under the influence of a potential energy field $V$.

While foundational, this equation does have boundaries:

1. Special Relativity: It assumes the particle is moving significantly slower than the speed of light. For particles moving at relativistic speeds, physicists must use the Dirac Equation.
2. Multiple Particles: For systems with multiple interacting particles (like heavy atoms), the formula remains conceptually similar, but the Hamiltonian operator ($\hat{H}$) becomes vastly more complex to account for particle interactions.
3. Spin: The original equation does not account for intrinsic quantum "spin." Modified versions, such as the Pauli equation, are required to incorporate this property.