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Chapter 34: Problem 7

How are the uncertainty principle and wave-particle duality related?

Short Answer

Expert verified
The uncertainty principle and wave-particle duality are connected in that the wave-particle duality leads to the formulation of the uncertainty principle. The more particle-like a quantum entity is (having a specific location), the less wave-like it is (having a specific momentum/wavelength), and vice versa, reflecting the impossibility of simultaneously knowing the exact position and momentum, as stated in the uncertainty principle.

Step by step solution

01

Understanding wave-particle duality

Wave-particle duality is the idea that every particle or quanta can exhibit behaviors indicative of not only particles, but also waves. This concept was first proposed by de Broglie in his hypothesis suggesting that any particle should be associated with a wave of wavelength \( \lambda \) = h/p, where h is Planck’s constant and p is the momentum of the particle.
02

Understanding the uncertainty principle

The Heisenberg's uncertainty principle states that it is impossible to simultaneously measure the exact position and momentum of a quantum particle. Mathematically, the principle is expressed as \( \Delta x \Delta p \geq \hbar / 2 \), where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( \hbar \) is the reduced Planck's constant (Planck's constant divided by 2\( \pi \)).
03

Interrelation between wave-particle duality and the uncertainty principle

Ideally, to fully know a 'particle', we would want to pin down both its exact location (particle-like property) and its momentum or wavelength (wave-like property). But according to wave-particle duality, the more a quantum object appears particle-like (i.e., located in a particular position), the less it appears wave-like, and vice versa. Thus, trying to measure both ends up violating the uncertainty principle, which states that both position and momentum cannot be simultaneously known to an arbitrary precision. This shows how the wave-particle duality idea led to the formulation of the uncertainty principle.

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