Mathematical Foundations for [Quantum Mechanics and Quantum Computation]() +##
[Predecessors of Quantum Mechanics](): Key Mathematicians and Their Contributions
-1- [Joseph Fourier](*) **(1822)**
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-* Developed the mathematical framework for the Fourier Transform, which is foundational in quantum mechanics and quantum computing.
- - Formula for Fourier Transform:
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- $\huge \color{DeepSkyBlue} \hat{f}(k) = \int_{-\infty}^{\infty} f(x) \, e^{-2\pi i k x} \, dx$
+1. **Leonhard Euler (1748)**
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-
+ * Developed the [Euler's Formula](), which links exponential functions to trigonometric functions. It is fundamental in wave mechanics and quantum state representation.
+ * **Euler's Formula:**
+ $\huge \color{DeepSkyBlue} e^{i\theta} = \cos(\theta) + i\sin(\theta)$
- - Formula for [Inverse]() Fourier Transform:
-
- $\huge \color{DeepSkyBlue} f(x) = \int_{-\infty}^{\infty} \hat{f}(k) \, e^{2\pi i k x} \, dk$
+ Where:
+ - **\( e \)**: Base of the natural logarithm.
+ - **\( \theta \)**: Phase angle.
+ - **\( i \)**: Imaginary unit.
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+ Euler's formula is essential for describing quantum wavefunctions and visualizing oscillations in the complex plane.
- [Where]():
- - $\large \color{DeepSkyBlue} f(x)$ is the original function in the spatial domain.
- - $\large \color{DeepSkyBlue} \hat{f}(k)$ is the transformed function in the frequency domain.
- - $\large \color{DeepSkyBlue} x$ represents position, and $k$ represents momentum or frequency.
+#
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- [**Relevance in Quantum Mechanics and Computing:**]()
- - **Quantum Mechanics**: Converts wavefunctions between position and momentum spaces.
- - **Quantum Computing**: Basis for the Quantum Fourier Transform (QFT), essential for algorithms like Shor's factoring algorithm.