I find quantum mechanics to be very hard, and I am currently having trobule with the following 2 problems, can someone please help me out? 1) A thin solid barrier in the xy-plane has a 10-micrometer-diameter circular hole. An electron traveling in the z-direction with v[sub]x[/sub]=0 m/s passes through the hole. Afterward, is v[sub]x[/sub] still zero? If not, within what range is v[sub]x[/sub] likely to be? My textbook says that the uncertainty principle is (delta x)(delta p[sub]x[/sub]) > h/2 But I am not sure what delta x would be in this situation...and I got completely confused with the x,y,z direction business as well...if v[sub]x[/sub] is zero originally, why would it possibly be different after going through the hole? I don't understand what is going on... 2) A typical electron in a piece of metallic sodium has energy -E[sub]o[/sub] compared to a free electron, where E[sub]o[/sub] is the 2.7 eV work function of sodium. At what distance beyond the surface of the metal is the electron's probability density 10% of its value at the surface? First of all, I don't get what it means by saying that the elctron has energy "-E[sub]o[/sub] compared to a free electron", what is the energy of a free electron? why is there a negative sign in front of E[sub]o[/sub]? Secondly, I don't even know which equation to use, can someone give me some hints on solving this problem? I am totally lost... Any help is greatly appreciated!Please Register or Log in to view the hidden image!
It's only hard if you think about it. Try not thinking and just doing the calculations. This comment will, of course, be taken out of context by the following people: Farsight Singularity MetaKraon zanket (perhaps) If any of these people post replies in this thread do not listen to them.
Defining energies is always relative. This is analagous to the problems from your physics one class where you had to pick a zero for potential energy. Where is the potential energy zero? Wherever you want it to be. It seems that the free electrons have been defined to have zero energy, and the electrons stuck to the sodium atoms have a lower energy (i.e. they sit at a lower level) than the other electrons. Let me try to make a picture: ------------- 0 eV ------------------------ -2.7 eV
For 2 it looks like you should have a probability distribution function for the electron, something like \(N e^{-\sigma x^2}\) or something. (This equation may be wrong!) Check you text.
For 1: http://en.wikipedia.org/wiki/Double-slit_experiment Wikipedia is a great resourse for things like this. The electron has properties of a wave that make it behave like a wave in some scattering experiments. I hope this few comments were helpful. I'm sure others can explain betterPlease Register or Log in to view the hidden image!
1) The electron is moving in the z-direction only and we are given that v[sub]x[/sub]=0 m/s, so v[sub]x[/sub] should be 0 m/s always, right? Passing a hole shouldn't affect anything...there is no force acting on it...
For #2, I can't find your formula in my textbook... Can someone guide me through the approach to solving this problem? I just don't know how to begin...
1) While the electron is in the hole, we know its position in the x direction to a delta x of 10 micrometer, yes? So what does this say about the uncertainty in the x component of momentum?
winner--- Ignore what I said about finding the probability distribution. It seems that you need to use the uncertainty principle to find #2. You want to find x such that the uncertainty in measuring the position is 10%. This means, for you \(\Delta x = 0.10 x\). I haven't been able to think about how to find \(\Delta p\) from \(KE = hf - \phi\), but can tell you that you may need something like \(KE = \frac{p^2}{2m}\). The answer is floating around here somewhere.
1) The only way I can match the answer is to substitute delta x=10x10^-6 m, why is that? Is it because it may be at one end?
It can be at either edge of the hole, so the diameter of the hole is the uncertainty in x. The point is that as the electron passes through the hole it is constrained to be withing +/- 5 micrometer of the axis, both in x and y coordinants. So delta x and delta y are both equal to 10 micrometer. This implies an uncertainty in both the x and y components of the momemtum.
