Understanding the Uncertainty Principle

Understanding the Uncertainty Principle: Combining Wave and Matrix Mechanics

Abstract

This paper explains the Heisenberg uncertainty principle, focusing on the position-momentum rule $\Delta x\Delta p\ge\hbar/2,$ using two key quantum mechanics ideas: wave mechanics and matrix mechanics. Wave mechanics uses Fourier transforms to show how a particle's position and momentum are linked. Matrix mechanics uses math with operators to focus on the special relationship between position and momentum operators.

Keywords: Heisenberg uncertainty principle, wave mechanics, Fourier transform, matrix mechanics, operator relation

1 Introduction

1.1 Background of the Uncertainty Principle

In 1927, Werner Heisenberg introduced the uncertainty principle, saying you can't measure things like position and momentum exactly at the same time. He used thought experiments, like the gamma-ray microscope, to show that measuring one affects the other, initially suggesting $\Delta x\Delta p\sim h.$ Later, Earle Hesse Kennard (1927) and Hermann Weyl (1928) gave clearer math proofs showing $\Delta x\Delta p\ge\hbar/2,$ proving this uncertainty is a built-in part of quantum systems, not just from measuring.

1.2 Two Ways to Look at Quantum Mechanics

In 1926, Erwin Schrödinger created wave mechanics, using Louis de Broglie's idea that particles act like waves, describing them with wave functions $\psi(x,t)$. Around the same time, in 1925, Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics, using matrices to represent measurements. Schrödinger showed in 1926 that these two methods are mathematically the same.

2 Uncertainty Principle in Wave Mechanics

2.1 Wave Functions for Position and Momentum

In wave mechanics, a particle is described by a position wave function $\psi(x)$ or a momentum wave function $\phi(p)$. The chance of finding a particle near position x in a small range dx is $|\psi(x)|^{2}dx,$ with the total chance adding up to 1:

$$\int_{-\infty}^{\infty}|\psi(x)|^{2}dx=1$$

Similarly, the chance of a particle having momentum near p in a range is $|\phi(p)|^{2}$, with:

$$\int_{-\infty}^{\infty}|\phi(p)|^{2}dp=1$$

2.2 How Fourier Transforms Connect Them

The wave functions $\psi(x)$ and $\phi(p)$ are linked by Fourier transforms:

$$\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x)e^{-ipx/\hbar}dx$$

$$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\phi(p)e^{ipx/\hbar}dp$$

If a wave function is very narrow in position (x-space), its Fourier transform spreads out in momentum (p-space), and vice versa. This is the math behind the uncertainty principle in wave mechanics.

2.3 What Uncertainty Means: Standard Deviation

The uncertainty (or spread) $\Delta A$ of a measurement A is:

$$(\Delta A)^{2}=\langle A^{2}\rangle-\langle A\rangle^{2},$$

where $\langle A\rangle=\int_{-\infty}^{\infty}\psi^{*}(x)A\psi(x)dx.$ For position, the spread is:

$$(\Delta x)^{2}=\int_{-\infty}^{\infty}x^{2}|\psi(x)|^{2}dx-(\int_{-\infty}^{\infty}x|\psi(x)|^{2}dx)^{2}$$

For momentum, using the momentum operator $\hat{p}=-i\hbar\frac{d}{dx}$, the spread is:

$$(\Delta p)^{2}=\langle\hat{p}^{2}\rangle-\langle\hat{p}\rangle^{2}.$$

If $\langle p\rangle=0,$ this becomes:

$$(\Delta p)^{2}=\int_{-\infty}^{\infty}\psi^{*}(x)(-\hbar^{2}\frac{d^{2}}{dx^{2}})\psi(x)dx=\hbar^{2}\int_{-\infty}^{\infty}|\frac{d\psi(x)}{dx}|^{2}dx$$

2.4 Proving $\Delta x\Delta p\ge\hbar/2$ (Bandwidth Rule)

To make it simpler, assume $\langle x\rangle=0$ and $\langle p\rangle=0$ by shifting coordinates, which doesn't change the uncertainty. For a normalized function $f(x)$ and its Fourier transform $F(k)$ (where k is wave number), the Cauchy-Schwarz inequality gives:

$$\Delta x\Delta k\ge\frac{1}{2}.$$

Using de Broglie's relation $p=\hbar k$ we get $\Delta p=\hbar\Delta k.$ Plugging in:

$$\Delta x(\frac{\Delta p}{\hbar})\ge\frac{1}{2}.$$

Multiply by $\hbar:$

$$\Delta x\Delta p\ge\frac{\hbar}{2}.$$

A Gaussian wave packet, where $\Delta x\Delta p=\hbar/2$, is an example of the smallest possible uncertainty.

3 Uncertainty Principle in Matrix Mechanics

3.1 Measurements as Operators and States as Vectors

In matrix mechanics, measurements are represented by Hermitian operators (like position $\hat{Q}$ and momentum $\hat{P}$) acting on state vectors $|\psi\rangle$ in a math space called Hilbert space.

3.2 Special Operator Rule (CCR)

Position and momentum operators follow:

$$[\hat{Q},\hat{P}]=\hat{Q}\hat{P}-\hat{P}\hat{Q}=i\hbar\hat{I},$$

where $\hat{I}$ is the identity operator. This non-swapping property is key to quantum mechanics.

3.3 General Uncertainty Rule (Robertson-Schrödinger)

For any two Hermitian operators $\hat{A}$ and $\hat{B}$, the Robertson inequality says:

$$(\Delta A)^{2}(\Delta B)^{2}\ge(\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle)^{2}.$$

Define $\hat{A}^{\prime}=\hat{A}-\langle\hat{A}\rangle\hat{I}$ and $\hat{B^{\prime}}=\hat{B}-\langle\hat{B}\rangle\hat{I}.$ Let $|f\rangle=\hat{A}^{\prime}|\psi\rangle$ and $|g\rangle=\hat{B}^{\prime}|\psi\rangle$. The Cauchy-Schwarz inequality $\langle f|f\rangle\langle g|g\rangle\ge|\langle f|g\rangle|^{2}$ means:

$$(\Delta A)^{2}(\Delta B)^{2}\ge|\langle\hat{A}^{\prime}\hat{B}^{\prime}\rangle|^{2}.$$

Since $\hat{A}^{\prime}\hat{B}^{\prime}=\frac{1}{2}[\hat{A}^{\prime},\hat{B}^{\prime}]+\frac{1}{2}\{\hat{A}^{\prime},\hat{B}^{\prime}\},$ where $[\hat{A}^{\prime},\hat{B}^{\prime}]$ gives imaginary results and $\{\hat{A}^{\prime},\hat{B}^{\prime}\}$ gives real ones, we get:

$$\langle\hat{A}^{\prime}\hat{B}^{\prime}\rangle=\frac{1}{2}\langle[\hat{A}^{\prime},\hat{B}^{\prime}]\rangle+\frac{1}{2}\langle\{\hat{A}^{\prime},\hat{B}^{\prime}\}\rangle.$$

So:

$$|\langle\hat{A}^{\prime}\hat{B}^{\prime}\rangle|^{2}=|\frac{1}{2}\langle[\hat{A}^{\prime},\hat{B}^{\prime}]\rangle|^{2}+|\frac{1}{2}\langle\{\hat{A}^{\prime},\hat{B}^{\prime}\}\rangle|^{2}.$$

Since $[\hat{A}^{\prime},\hat{B}^{\prime}]=[\hat{A},\hat{B}]$, and for imaginary $\langle[\hat{A}^{\prime},\hat{B}^{\prime}]\rangle=iC$ (C real), $|\frac{1}{2}iC|^{2}=(\frac{c}{2})^{2},$ matching $(\frac{1}{2i}iC)^{2}.$

Thus:

$$|\langle\hat{A}^{\prime}\hat{B}^{\prime}\rangle|^{2}=(\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle)^{2}+(\frac{1}{2}\langle\{\hat{A}^{\prime},\hat{B}^{\prime}\}\rangle)^{2}.$$

This gives the Schrödinger uncertainty rule:

$$(\Delta A)^{2}(\Delta B)^{2}\ge(\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle)^{2}+(\frac{1}{2}\langle\{\hat{A}^{\prime},\hat{B}^{\prime}\}\rangle)^{2}.$$

The Robertson rule comes from the second term being non-negative.

3.4 Proving $\Delta x\Delta p\ge\hbar/2$ from CCR

Set $\hat{A}=\hat{Q}$ and $\hat{B}=\hat{P}.$ Given $[\hat{Q},\hat{P}]=i\hbar\hat{I}$

$$\langle[\hat{Q},\hat{P}]\rangle=\langle i\hbar\hat{I}\rangle=i\hbar.$$

Plug into the Robertson inequality:

$$(\Delta Q)^{2}(\Delta P)^{2}\ge(\frac{1}{2i}(i\hbar))^{2}=(\frac{\hbar}{2})^{2}.$$

Take the square root (spreads are non-negative):

$$\Delta Q\Delta P\ge\frac{\hbar}{2}.$$

So, $\Delta x\Delta p \ge \hbar/2$.

4 Discussion

4.1 Comparing Wave and Matrix Mechanics

Wave mechanics clearly shows how position and momentum spreads are tied together using Fourier transforms. Matrix mechanics uses the math of operators not swapping to give a more general proof. Both methods reach $\Delta x\Delta p\ge\hbar/2$, showing quantum mechanics is consistent.

5 Conclusion

5.1 What We Found

This paper proved the Heisenberg uncertainty principle $\Delta x\Delta p\ge\hbar/2$ using Fourier transforms in wave mechanics and operator math in matrix mechanics. Both show this principle is a core part of quantum mechanics.

5.2 Why the Uncertainty Principle Matters

The Heisenberg uncertainty principle is a key idea in quantum mechanics, breaking from classical physics' predictable view. It's not a flaw but a building block that supports how matter stays stable and quantum effects work.

References

Wikipedia, "Uncertainty Principle Overview"
Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen," Zeitschrift für Physik, 44(4-5), 326-352.
Weyl, H. (1928). Gruppentheorie und Quantenmechanik. Leipzig: S. Hirzel.
Wikipedia, "Fourier Transform and Uncertainty Principle"
Wikipedia, "Schrödinger Equation"
Wikipedia, "Matrix Mechanics"
Wikipedia, "Canonical Commutation Relation"
Wikipedia, "Robertson-Schrödinger Uncertainty Relation"
Wikipedia, "Cauchy-Schwarz Inequality"