In this post,I intend to write something on the position and momentum measurement of some particle in the light of uncertainty principle.Prototype example is the famous thought experiment by Heisenberg [Gamma ray microscope].I will show the picture from Scwabl's QM book p-21.
The text I am following now is written by Gottfried.It's a rather insightful book.In the connection of position and momentum measurement,the concept of Compton effect turns out to be so helpful.
I am quoting some parts from the text which looks interesting to me and I think you will also be interested.I thought of this as follows:
In case of Compton effect,if we use light of wavelength much below than λ_C,the change in the wavelength is still of the order of λ_C,and thus,λ'~λ_C; which means much of the energy of the incoming photon has been imparted to the electron.To reveal the details,or,to increase resolution,we have disturbed our system appreciably which is undesirable.So,we need to have a balance of effective minimum and meaningful maximum of the wavelength which is precisely of the order of λ_C.Hence,to have better resolution,we may decrease λ,but not beyond this limit.
Resolution R=λ/sin(φ) where θ=(90-φ) is the lab scattering angle for the photon.As photon scattering angle θ--->0, φ--->90.This happens when the photon is scattered away from the lens [fig: Franz Schwabl,QM-p21] and will not enter the microscope.This also means λ~λ'.As a result, the microscope is flooded with too much ineffective light.Precisely, this happens for longer wavelength limit: (δλ/λ) x 100%
As λ_C--->0 in the non-relativistic limit, we are able to specify the particle's position with fantastic accuracy.
Any suggestion to the betterment will be gladly appreciated.
-Kolahal
The text I am following now is written by Gottfried.It's a rather insightful book.In the connection of position and momentum measurement,the concept of Compton effect turns out to be so helpful.
I am quoting some parts from the text which looks interesting to me and I think you will also be interested.I thought of this as follows:
In case of Compton effect,if we use light of wavelength much below than λ_C,the change in the wavelength is still of the order of λ_C,and thus,λ'~λ_C; which means much of the energy of the incoming photon has been imparted to the electron.To reveal the details,or,to increase resolution,we have disturbed our system appreciably which is undesirable.So,we need to have a balance of effective minimum and meaningful maximum of the wavelength which is precisely of the order of λ_C.Hence,to have better resolution,we may decrease λ,but not beyond this limit.
Resolution R=λ/sin(φ) where θ=(90-φ) is the lab scattering angle for the photon.As photon scattering angle θ--->0, φ--->90.This happens when the photon is scattered away from the lens [fig: Franz Schwabl,QM-p21] and will not enter the microscope.This also means λ~λ'.As a result, the microscope is flooded with too much ineffective light.Precisely, this happens for longer wavelength limit: (δλ/λ) x 100%
Since,Resolution R=λ/sinΦ,hence, dR=dλ/sinΦ~λC/sinΦ~λC.
As λ_C--->0 in the non-relativistic limit, we are able to specify the particle's position with fantastic accuracy.
Any suggestion to the betterment will be gladly appreciated.
-Kolahal
