Bridging Classical Circuits and Quantum Mechanics: From Capacitors to Planck’s Constant

Using Fourier Transforms, Euler’s Formula, and Planck’s Constant to Connect Circuit Theory and the Quantum World

Abstract

This article explores the mathematical and conceptual connections between classical circuit theory and quantum mechanics, beginning with the capacitor’s current-voltage relationship I(t) = C * dV(t)/dt. Through Fourier transforms and Euler’s formula, we analyze its frequency-domain behavior and further link it to Planck’s constant ℏ in quantum mechanics, which serves as a metronome for energy-frequency conversion. Using a reservoir analogy, we intuitively explain these concepts and employ visualizations to highlight the contrast between classical frequency and quantum energy. This article aims to provide a unified perspective for interdisciplinary research, demonstrating how mathematical tools bridge the classical and quantum worlds.

1. Introduction

In classical circuit theory, the capacitor is a fundamental passive component, with its current-voltage relationship described by the differential equation:

𝐼(𝑡) = 𝐶 · 𝑑𝑉(𝑡)/𝑑𝑡

This equation reveals the dynamic relationship between current and voltage and forms the basis for frequency-domain analysis and phase behavior. Through Fourier transforms and Euler’s formula, we can convert time-domain signals into the frequency domain, uncovering the characteristic that current leads voltage by 90°. These mathematical tools are not only applicable to circuit analysis but also have profound connections with wave functions and energy quantization in quantum mechanics. Planck’s constant ℏ converts frequency ω into energy E via E = ℏ * ω, acting as the “metronome” of the quantum world. This article elucidates the unity of these concepts through mathematical derivations, intuitive analogies, and visualizations.

2. Capacitor Current-Voltage Relationship

 2.1 Time-Domain Representation

The capacitor’s current-voltage relationship is described by:

𝐼(𝑡) = 𝐶 · 𝑑𝑉(𝑡)/𝑑𝑡

Where:

- I(t): Current (Amperes, A)

- C: Capacitance (Farads, F)

- V(t): Voltage (Volts, V)

- dV(t)/dt: Rate of change of voltage over time

This equation indicates that current is proportional to the rate of change of voltage, with capacitance C determining the sensitivity of current to voltage changes.

 2.2 Reservoir Analogy

To intuitively understand this relationship, consider a reservoir analogy:

- Voltage V(t): Water level in the reservoir, varying over time.

- Current I(t): Rate of water flow into or out of the reservoir.

- Capacitance C: Width of the reservoir (storage capacity).

A wide reservoir (large C) requires a larger water flow (current) to change the same water level (voltage), while a narrow reservoir (small C) is more sensitive to water level changes. This reflects the amplifying role of C in the equation.

 3. Fourier Transform: From Time Domain to Frequency Domain

 3.1 Definition of Fourier Transform

The Fourier transform decomposes time-domain signals into frequency-domain sinusoidal components. For voltage V(t), the Fourier transform is defined as:

𝑉(𝜔) = ∫₋∞^∞ 𝑉(𝑡) · 𝑒^(⁻ʲ𝜔𝑡) 𝑑𝑡

Where:

- ω: Angular frequency (radians/second)

- j: Imaginary unit, j^2 = -1

- e^(-jωt): Exponential form from Euler’s formula

The Fourier transform of the time derivative has an important property:

ℱ{𝑑𝑉(𝑡)/𝑑𝑡} = 𝑗𝜔𝑉(𝜔)

 3.2 Capacitor Behavior in Frequency Domain

Applying the Fourier transform to I(t) = C * dV(t)/dt, we obtain:

𝐼(𝜔) = 𝐶 · 𝑗𝜔𝑉(𝜔)

This shows that current I(ω) is proportional to voltage V(ω), with the proportionality factor jωC incorporating frequency ω and phase factor j. The impedance of the capacitor is defined as:

𝑍꜀(𝜔) = 𝑉(𝜔)/𝐼(𝜔) = 1/(𝑗𝜔𝐶)

 3.3 Frequency Domain in Reservoir Analogy

In the reservoir analogy, complex water level fluctuations V(t) resemble waves on the sea surface, which can be decomposed into single-frequency sine waves V(ω). The Fourier transform acts as a “wave decomposer,” separating each frequency component. The frequency-domain equation I(ω) = jωCV(ω) indicates that for each frequency wave, the water flow rate (current) depends on wave frequency ω, capacitance C, and wave amplitude V(ω).

 4. Euler’s Formula: Phase and Rhythm

 4.1 Euler’s Formula

Euler’s formula connects complex exponentials with trigonometric functions:

𝑒^(𝑗𝜃) = cos(𝜃) + 𝑗·sin(𝜃)

In capacitor impedance, 1/j = -j, and according to Euler’s formula:

1/𝑗 = 𝑒^(⁻𝑗𝜋⁄2)

Thus, impedance can be expressed as:

𝑍꜀(𝜔) = 1/(𝑗𝜔𝐶) = (1⁄(𝜔𝐶)) · 𝑒^(⁻𝑗𝜋⁄2)

This indicates that the magnitude of impedance is 1/(ωC), with a phase of -π/2 (-90°), meaning current leads voltage by 90°.

4.2 Phase Difference in Reservoir

In the reservoir analogy, when the water level wave is “ascending” (V(t) increasing, dV(t)/dt > 0), the water flow (current) rapidly enters the reservoir, reaching its maximum. Euler’s formula quantifies this “rhythm difference”: the current waveform leads the voltage waveform by π/2 radians. For example, if V(t) = cos(ωt), then:

𝐼(𝑡) = 𝐶 · 𝜔 · sin(𝜔𝑡) = 𝐶 · 𝜔 · cos(𝜔𝑡 + 𝜋⁄2)

This shows that the current waveform leads the voltage by 90°, similar to water flow peaking before the water level crest.

 4.3 Phase of Quantum Wave Function

Euler’s formula is equally important in quantum mechanics. Quantum wave functions are often expressed as e^(j(kx - ωt)), with phase controlled by ωt, related to Planck’s constant ℏ through E = ℏω. This phase rotation is analogous to the current-voltage rhythm difference in circuits, demonstrating mathematical unity.

 5. Planck’s Constant: Quantum Metronome

5.1 Energy-Frequency Relationship

In quantum mechanics, Planck’s constant ℏ converts frequency ω into energy:

E = ℏ * ω

Where:

- ℏ ≈ 1.0545718 × 10^(-34) J·s (reduced Planck’s constant)

- E: Energy (Joules, J)

- ω: Angular frequency (radians/second)

This equation shows that frequency ω determines the energy levels of quantum systems, with ℏ as the proportional constant, acting as the “metronome” controlling the rhythm of energy quantization.

5.2 Quantum Reservoir Analogy

Extending the reservoir analogy to the quantum world:

- Classical reservoir: Water level (voltage) changes continuously, water flow (current) varies with water level rate of change.

- Quantum reservoir: Water level is quantized into discrete energy levels (E_n = n * ℏ * ω), water flow (quantum state transitions) can only jump between these discrete levels.

Planck’s constant ℏ determines the spacing between energy levels, similar to how capacitance C determines the response of current to voltage changes.

 5.3 Classical vs. Quantum Comparison

- Classical circuits: Frequency ω determines waveform oscillation speed, such as sine wave behavior of voltage and current.

- Quantum world: Frequency ω determines discrete energy levels via E = ℏ * ω, such as quantum states of photons or electrons.

Fourier transforms decompose classical waveforms into frequency components, analogous to energy levels corresponding to frequencies in quantum systems, showing their mathematical connection.

6. Visualization: Bridging Classical and Quantum

To intuitively demonstrate these concepts, we designed the following visualizations:

6.1 Figure 1: Voltage and Current Waveforms

- Content: Plot voltage V(t) = cos(ωt) and current 𝐼(𝑡) = 𝐶 · 𝜔 · sin(𝜔𝑡)

, assuming ω = 1, C = 1.

- Features: Current (red dashed line) leads voltage (blue solid line) by 90°, mark “wave ascending” points to show maximum water flow when water level rises.

- Analogy: Side text explains “rhythm difference between water level (voltage) and water flow (current)”.

6.2 Figure 4: Planck’s Constant as Metronome

- Content: Plot E = ℏ * ω for angular frequency ω ranging from 0 to 10^15 rad/s, and mark specific frequencies (for example, 10^13 rad/s and 5 × 10^13 rad/s) with discrete energy points.

- Features:

  - A blue line representing the continuous relation E = ℏ * ω.

  - Red dots indicating the quantized energy levels 𝐸ₙ = ℏ · 𝜔ₙ

  - Annotation: “ℏ converts frequency into energy.”

- Analogy: A “quantum reservoir” illustration showing discrete energy levels as quantized water-levels.

6.3 Integrated Figure

- Layout:

  - Top left: Figure 1 showing the voltage–current phase shift.

  - Top right: Figure 4 showing the energy–frequency relation.

  - Bottom: A textual comparison of the classical case (frequency determines waveform) versus the quantum case (frequency determines energy levels).

- Purpose: Emphasize the different roles of angular frequency ω in classical and quantum systems, with ℏ acting as the “quantum metronome.”

7. Discussion

Through the capacitor’s current–voltage relation 𝐼(𝑡) = 𝐶 · 𝑑𝑉(𝑡)/𝑑𝑡 , we have shown how Fourier transforms and Euler’s formula convert time-domain analysis into the frequency domain and reveal the phase relationship (current leads voltage by 90°). The reservoir analogy makes these concepts more intuitive: capacitance C is like reservoir width, frequency ω sets the wave speed, and the phase shift reflects the “rhythm” difference between flow and level. These mathematical tools extend seamlessly into quantum mechanics: Planck’s constant ℏ converts frequency ω into energy E, analogous to how C scales current sensitivity. Our visualizations reinforce this unity by showing clearly the role of ω in both classical and quantum regimes.

Future work could explore analogies for other circuit elements (such as inductors or resistors) with quantum systems, or delve deeper into applying Fourier analysis to quantum wave-function decomposition. Such interdisciplinary links not only deepen our understanding of physical systems but also provide theoretical foundations for emerging technologies like quantum computing and nanoelectronics.

 8. Conclusion

The capacitor’s relation 𝐼(𝑡) = 𝐶 · 𝑑𝑉(𝑡)/𝑑𝑡

 together with Fourier transforms and Euler’s formula, highlights the central roles of frequency and phase in classical circuits. Planck’s constant ℏ extends these ideas into the quantum world via E = ℏ * ω. In the frequency domain we see the spectrum of voltage changes as modes in a field theory “concert,” Euler’s formula translates phase rotations into computable geometry, and ℏ quietly sets the energy of each “beat.” Thus, a capacitor does more than respond to present voltage—it senses the trend toward future change. This sensitivity to change itself is the essence of a quantum field. Far from a static backdrop, quantum fields are dynamic “seas of energy,” and superconducting LC circuits used in modern quantum devices are miniature simulators of that sea. From transmon-qubit level structures to vacuum-photon excitation in microwave cavities, today’s quantum technologies fuse capacitors, inductors, and nonlinear elements into operable quantum-field simulators.

Remarkably, these devices have transcended engineering roles and evolved into platforms for probing fundamental physics—simulating post-inflation vacuum fluctuations or black-hole–like particle creation. This trend underscores an emerging view: circuits are not merely conduits for power but arenas for “analog computation” of nature’s laws. A capacitor’s invention, then, is not only a tool for smoothing and filtering; it may be an origin point for our understanding of the universe. The tools of signal processing thus become gateways to deeper quantum exploration. As engineering and physics converge, circuit designers may become the next generation of field-theory architects. In this sense, capacitance is not just energy storage but the resonator of rhythm, field, and cosmic structure—building systems that echo change and existence itself.

Related Topic: Circuit Field Theory: Using Electrical Circuits as a Cross-Scale Modeling Platform for Quantum Systems

https://simonchou.blogspot.com/2025/05/classical-circuits-and-quantum.html

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Visualization Details and Python Script

Plot 1: Voltage and Current Waveforms

• Show voltage: 𝑉(𝑡) = cos(ω𝑡) and current: 𝐼(𝑡) = 𝐶ω sin(ω𝑡)

• Highlight that current leads voltage by 90° in a capacitor

• Add clear annotations to explain this phase lead

• Include an inset: a sine wave represents water level (𝑉), and an arrow represents flow (𝐼) — a reservoir analogy

• Link this analogy to Plot 3

Plot 4: Planck–Einstein Relation

• Plot: 𝐸 = ℏω (energy vs. frequency)

• Add inset analogy:

  • Energy = water level

  • Frequency = flow rate

  • Represent quantum “steps” (discrete levels), versus classical continuity

• Include secondary axis for photon wavelength (λ)

• Label ℏ as reduced Planck’s constant

Plot 2: Fourier Transform of a Square Wave

• Show time-domain square wave and its frequency-domain Fourier series

• Emphasize that only odd harmonics (1st, 3rd, 5th, …) are present

• Improve axis labels and harmonic markers

• Add a brief note: frequency components relate to energy levels (see Plot 4) in quantum systems

Plot 3: Euler’s Formula and Quantum Phase Animation

• Display unit circle with complex exponential: 𝑒^{𝑗θ} = cos(θ) + 𝑗 sin(θ)

• Show vectors:

  • Voltage → cos(ω𝑡)

  • Current → sin(ω𝑡)

  • Quantum phase → vector starting at 45°

• Add inset: sine wave (reservoir level), arrow (flow), to match Plot 1 analogy

• Animate the quantum vector's rotation (phase evolution), save as GIF

• Annotate:

  • Euler’s formula

  • Phase lead between current and voltage

  • Quantum phase dynamics in simple terms

References

1. Oppenheim, A. V., & Schafer, R. W. (2010). *Discrete-Time Signal Processing*. Prentice Hall.

2. Griffiths, D. J. (2005). *Introduction to Quantum Mechanics*. Pearson Education.

3. Boylestad, R. L. (2010). *Introductory Circuit Analysis*. Prentice Hall.

4. https://github.com/fullyloaded/Python-Code