Equation for the resolving power of a microscope?

[Generic Turtle]

Hi I'm reading through a Quantum Mechanics textbook called Quantum Mechanics by Book by Alastair I. M. Rae and in the opening chapter it talks about the Heisenberg uncertainty principle and talks about how a measurement of position of a particle causes an uncertainty from the momentum due to the recoil from the interaction with the illuminating photon.

In the mathematical analysis is sates that in standard optical theory the uncertainty in the resolving uncertainty for the position of an object is.

Δx≈λ/sin(α)

I've tried googling various combinations of the words lens, microscope, resolution, and resolving power and I have not found where this equation/approximation comes from. Can anyone please be of a help?

 

[Drakkith]

in the opening chapter it talks about the Heisenberg uncertainty principle and talks about how a measurement of position of a particle causes an uncertainty from the momentum due to the recoil from the interaction with the illuminating photon.

That's a terrible way to introduce the uncertainty principle, because the Heisenberg Uncertainty Principle (HUP) has nothing to do with the effects of observation. That's a different effect known as the Observer Effect. The HUP does not arise because of any uncertainty with the recoil of the particle from something like a photon used in a microscope. Instead it arises because particles are fundamentally represented by wave functions, which have certain properties that cannot be simultaneously known to any arbitrary accuracy.

Classically you can see this when you try to pin down the location and frequency of a wave. (See this article: http://www.mtnmath.com/whatrh/node72.html#FigSine)
A pure frequency wave extends forever in all directions, and so literally has no one specific location. No real waves exist as a single pure frequency, instead they all exist as the sum of multiple frequencies, narrowing their possible locations in space. This relationship between position and frequency is analogous to the HUP. You can't have a wave that exists in only one spot without having it consist of an infinite number of frequencies. Another way of saying this is that constraining the position of a wave, perhaps by putting it in a box, requires that the wave be made out of many frequencies instead of one.

Of course, unlike particles in quantum mechanics, classical waves are continuous and so you can often find the wave a many different locations at the same time. Just an example of how classical and quantum mechanics differ.

In quantum mechanics particles have wave functions that give the probability of finding the particle in a specific position with a specific momentum (and other properties). True to their name, wave functions are, well, waves (Or at least they mathematically resemble classical waves). As such, they follow the same types of rules as a classical wave. This gives rise to properties that are 'counter' to each other, known in QM as being non-commutative. In QM position and momentum are non-commutative (time and energy are non-commutative as well).

One other thing. In the original language in which the principle was first explained (German I believe), this effect was called something like 'indeterminacy'. It was translated to English as 'uncertainty', much to the detriment of students everywhere.

 

[Cutter Ketch]

I've tried googling various combinations of the words lens, microscope, resolution, and resolving power and I have not found where this equation/approximation comes from. Can anyone please be of a help?

The phrase you are looking for is “diffraction limit”

 

[Tom.G]

resolving uncertainty for the position of an object

Not as accurate as @Drakkith put it, but in everyday words consider this:

• You are somewhat near a roadway and you want to know the position of a particular car on the road and how fast it is going as it passes you.
• You can measure or estimate the time it takes to travel a given distance, and you can measure or estimate its position.
• If you estimate its speed you can not know its exact position because it has moved while you where measuring the speed.
• Also you can estimate its position at any instant but that will not give you its speed because speed = distance/ time and an instant is zero elapsed time.

Cheers,
Tom