DAY 6: My learning #Quantum30 Challenge
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| Photo by Fractal Hassan on Unsplash |
Introduction:
Quantum mechanics revolutionized our understanding of the physical world, introducing the idea that fundamental properties like position and momentum have inherent limits to their precision due to the Heisenberg Uncertainty Principle.
Uncertainty and Conjugate Variables:
The Heisenberg Uncertainty Principle asserts that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is always greater than or equal to a constant (h-bar / 2), where h-bar is the reduced Planck's constant. Mathematically, this is expressed as:
Δx * Δp ≥ h-bar / 2
This principle suggests that the more accurately we know a particle's position, the less accurately we can know its momentum, and vice versa. Hence, position and momentum are considered conjugate variables in quantum mechanics.
Relationship to Fourier Transform:
The concept of conjugate variables in quantum mechanics is analogous to the Fourier Transform in signal processing. The Fourier Transform relates a function in the time domain to its representation in the frequency domain. In essence, it decomposes a complex signal into its constituent sinusoidal components. This relationship can be illustrated through examples:
Fourier Transform Example 1: Audio Signal Processing
Consider an audio signal representing a musical note. In the time domain, the waveform provides information about the note's amplitude and duration (analogous to position in quantum mechanics). When we apply a Fourier Transform, we obtain a frequency spectrum that reveals the constituent harmonics (analogous to momentum). A narrow spike in the frequency spectrum corresponds to a well-defined pitch (precise frequency) but lacks information about the note's precise starting and ending times.
Consider this example. A single sound wave we emit is a mixture of multiple sine wave of different frequency
Our test waveform = Summation i runs from 0 to N { (No. of waves[i]) x (wave of frequency[i] }
So, our Fourier transform is the plot of no. of waves in Y axis and frequency of waves in X axis.
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| Example for Fourier Transform Plot |
Conclusion:
The Heisenberg Uncertainty Principle and the concept of conjugate variables in quantum mechanics demonstrate that there are inherent limits to our ability to simultaneously know certain pairs of properties. This concept bears a resemblance to the Fourier Transform in signal processing, where precise knowledge in one domain implies uncertainty in the other. Understanding these principles is fundamental to both quantum mechanics and signal processing, with applications in various scientific and engineering fields.