The inequality in the Heisenberg uncertainty relation

In summary, the HUP (Heisenberg's Uncertainty Principle) is an inequality that specifies a minimum uncertainty across all states in quantum mechanics. While some states have an uncertainty equal to the minimum, most have a greater uncertainty. The reason for this inequality is that when analyzing a wave packet, the spatial frequency spectral width is inversely proportional to the spatial width. Although some have initially thought that this was due to squaring everything to get probability distributions, this is not the case. Additionally, the use of Fock spaces is not necessary to understand this concept, as Schrödinger wave-mechanics is sufficient.
  • #1
Derek P
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I was musing about why the HUP is an inequality. If you analyse a wave packet the spatial frequency spectral width is inversely proportional to the spatial width. So there should be an equality such as Heisenberg's equation 3 in this paper. Has anyone got a simple explanation of where the inequality comes from? I initially thought it was the fact that we square everything to get probability distributions, but I seem to have a mental block so any help would be appreciated.

Should this be a B level question?
 
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  • #2
Any state has a variance for each observable. So, there is an equality there which you can calculate for any specific state.

The HUP, however, specifies a minimum across all states. Some states, e.g. Coherent states of the harmonic oscillator have an uncertainty equal to the minimum. Most states have an uncertainty greater than this.
 
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  • #3
Derek P said:
I was musing about why the HUP is an inequality. If you analyse a wave packet the spatial frequency spectral width is inversely proportional to the spatial width. So there should be an equality such as Heisenberg's equation 3 in this paper. Has anyone got a simple explanation of where the inequality comes from? I initially thought it was the fact that we square everything to get probability distributions, but I seem to have a mental block so any help would be appreciated.
Should this be a B level question?
One Indian meal later and mental block has gone. Forget it, thanks.
 
Last edited:
  • #4
You can also calculate the wave packets, for which the equality sign holds. That's a way to introduce coherent and squeezed states!
 
  • #5
vanhees71 said:
You can also calculate the wave packets, for which the equality sign holds. That's a way to introduce coherent and squeezed states!
Well, yes, once you have the infrastructure of Fock spaces in place :)
 
  • #6
You don't need Fock spaces here. Just Schrödinger wave-mechanics is sufficient.
 
  • #7
vanhees71 said:
You don't need Fock spaces here. Just Schrödinger wave-mechanics is sufficient.
Yes, you're right.
 

Related to The inequality in the Heisenberg uncertainty relation

1. What is the Heisenberg uncertainty relation?

The Heisenberg uncertainty relation is a fundamental principle in quantum mechanics that states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.

2. Why is the Heisenberg uncertainty relation important?

The Heisenberg uncertainty relation is important because it sets a fundamental limit on the accuracy of measurements in the quantum world. It also plays a key role in understanding the behavior of particles at the subatomic level.

3. What is the inequality in the Heisenberg uncertainty relation?

The inequality in the Heisenberg uncertainty relation is a mathematical expression that relates the uncertainties in two complementary physical properties, typically position and momentum. It states that the product of the uncertainties in these properties must always be greater than or equal to a certain value, known as Planck's constant divided by 2.

4. How does the Heisenberg uncertainty relation affect our daily lives?

The Heisenberg uncertainty relation has little impact on our daily lives as it primarily applies to the behavior of particles at the subatomic level. However, it has led to many technological advancements in fields such as computer science and medicine.

5. Are there any exceptions to the Heisenberg uncertainty relation?

There are no exceptions to the Heisenberg uncertainty relation as it is a fundamental principle in quantum mechanics. However, there are ways to minimize the uncertainty in certain measurements, such as using more precise instruments or cooling particles to reduce their movement.

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