Math of the uncertainty principle

In summary, the deltas of the uncertainty relation can be operated on a state |ψ> as <ψ|ΔA|ψ>. Physically, the Δ represents the variance of a distribution and gives a measure of how likely one is to deviate from the average value. The momentum probability distribution is the Fourier transform of the position probability distribution, and there is a strict one-to-one mapping between these two distributions. Each completely defines the other, making it a significant fact in quantum mechanics.
  • #1
Durandarte
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How can one operate the deltas of the uncertainty relation? I know they're supposed to be the standard dev, but how are they operated in physical reality? Is there some sort of function to make them have a physical meaning?
 
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  • #2
As you know (<ψ|ΔA|ψ>)2=<ψ|AA|ψ>-(<ψ|A|ψ>)2.
This is the way ΔA act on some state |ψ>.
 
  • #3
Physically the [itex]\Delta[/itex] represents the variance of a distribution (the width of the probability distribution). The variance gives a measure of how likely one is to deviate from the average value.
 
  • #4
Durandarte said:
How can one operate the deltas of the uncertainty relation? I know they're supposed to be the standard dev, but how are they operated in physical reality? Is there some sort of function to make them have a physical meaning?

I never much cared for this simple relation. I think it is much more interesting to know that the momentum probability distribution is the Fourier transform of the position probability distribution. If one distribution is spiky then the other is very broad. This is both more general and more informative, and avoids that word "certainty" which IMO mystifies the situation.

In other words, there is a strict one-to-one mapping between these two distributions. Each completely defines the other. If I were to pick one fact of quantum mechanics deserving of wider recognition, this would be it.
 

FAQ: Math of the uncertainty principle

1. What is the uncertainty principle in math?

The uncertainty principle is a mathematical concept that states the limitations of accurately measuring certain pairs of physical quantities, such as position and momentum, at the same time. It was first proposed by German physicist Werner Heisenberg in 1927.

2. How is the uncertainty principle calculated?

The uncertainty principle is calculated using the mathematical equation ΔxΔp ≥ h/4π, where Δx represents the uncertainty in position, Δp represents the uncertainty in momentum, and h is the Planck constant.

3. What does the uncertainty principle tell us about the nature of particles?

The uncertainty principle tells us that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This suggests that particles do not have a definite position or momentum, but instead exist as a wave of probabilities.

4. Can the uncertainty principle be violated?

According to our current understanding of quantum mechanics, the uncertainty principle cannot be violated. It is a fundamental principle that governs the behavior of particles at the subatomic level.

5. How does the uncertainty principle impact everyday life?

While the uncertainty principle may seem abstract and unrelated to our daily lives, it actually has practical applications in fields such as cryptography and computer science. It also plays a crucial role in technologies such as MRI machines and electron microscopes.

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