Quantum Theory 里的一个有趣事件 【第二章 #61： quantum entanglement】

kensai

原帖由 novidx 于 24-4-2008 02:35 AM 发表


我的意思是它们看上去好像是Independent的，但实际上应该不是，不然怎么样Inteference。

interference 这里不是指他有 interfere,
这里是指interference pattern,
只是一副图画罢了，


他这个jargon是很久以前,还没有发现quantum mechanics 时就用下来的术语，以前是指superposition of wave 照成interference pattern,



1905后，他们还是用这个term ' interference pattern' ，
在 wave theory 来讲，他们是wave interfere with each other,
但是在matters 来讲，其实是没有interfere each other 的。

看上来，实际上，也是independent event

就好像current 是 positive to negative 一样，以前的错误一样流传下来

[ 本帖最后由 kensai 于 24-4-2008 09:46 AM 编辑 ]

llo_ol

我認爲可以肯定的是：

１。每粒電子都是在經過雙夾縫的時候和　自己　干涉，而不其他因素

２。粒子經過雙夾縫的時候肯定顯現了波的性質

３。所有電子的行爲都不是孤立事件（也就是說，在數目非常多的時候，如果有Ｎ粒電子射到某條條紋上面，就幾乎會有Ｎ粒射到其他條紋上面）。如果每個電子的投射是孤立事件，那麽干涉條文照理就不會出現

新人王伏羲

原帖由 kensai 于 24-4-2008 01:51 AM 发表


每一粒电子都不知道而且不需要其他电子去了那里，
这个是independent event,

你玩骰子，每丢一次得到的号码都不知道前后得到的号码是多少，
但是你丢很多次很多次后，他们的机会率会自己平均成1/6,
但是骰 ...

那么kensai兄是讲那个比较多的电子的地方平均的几率比较高吗？可是为什么那边的几率会比较高呢？可以计算的吗？是不是那两个逢干扰到电子的momentum and position.
还有就是你这样讲就说电子也是物质，并不是波因为他们不是像波那样interfere,只是因为有uncertainty的出现才有那interference pattern.但是为什么用这个来讲wave-particle duality of matter呢？

[ 本帖最后由 新人王伏羲 于 24-4-2008 03:35 PM 编辑 ]

kensai

原帖由 新人王伏羲 于 24-4-2008 03:28 PM 发表

那么kensai兄是讲那个比较多的电子的地方平均的几率比较高吗？可是为什么那边的几率会比较高呢？可以计算的吗？是不是那两个逢干扰到电子的momentum and position.

对了，你讲对了重点，quantum mechanics 讲的就是机会率啊。
机会率可以用 intensity of  de broglie wave 来算。

我想表明一些东西，我之前讲没有interfere 是electrons 之间没有 interfere each other,independent
但是，electrons 和 slits 有interfere,这个才会造成interference pattern.

还有就是你这样讲就说电子也是物质，并不是波因为他们不是像波那样interfere,只是因为有uncertainty的出现才有那interference pattern.但是为什么用这个来讲wave-particle duality of matter呢？

wave-particle duality of matter 证实有 matter wave, 或者说当一个东西动时，他会放出wave, 而， lamda = h/p,他不动就没有wave
但是这种wave 不想其他wave 可以用仪器量出来，是很抽象的wave，但是act like normal wave

summary

motion-> debroglie wave-> interference pattern ( electrons interfere with slits , not with electrons)

kensai

The conclusion of all this is that there is no experiment that can tell us what the electrons are doing at the slits that does not also destroy the interference pattern. This seems to imply that there is no answer to the question of what is going on at the slits when we see the interference pattern. The path of the electron from the electron gun to the screen is not knowable when we see the interference pattern. As Heisenberg said, "The path [of the electron] comes into existence only when we observe it."

http://www.upscale.utoronto.ca/G ... lit/DoubleSlit.html
有耐性的话，就从The Two Slit Experiment for Electrons那里开始读起，
才两面罢了，不长。

kensai

从课本拿出来的

kensai

原帖由 新人王伏羲 于 24-4-2008 03:28 PM 发表

那么kensai兄是讲那个比较多的电子的地方平均的几率比较高吗？可是为什么那边的几率会比较高呢？可以计算的吗？是不是那两个逢干扰到电子的momentum and position.
还有就是你这样讲就说电子也是物质，并不是波因 ...

上次你问我的相对论的问题解决了没有？

新人王伏羲

那么电子在移动时是wave但他不像wave那样interference，而是没有interference的，每一粒电子都是independent的．电子和slit interfere 才会有interference pattern.
但是电子是如何和slit interfere 而产生interference pattern呢？是不是没有答案的．．．．．

至于那个相对论的问题，pole and barn paradox 我明白啦但是情况和我讲的问题有些不一样。所以还是不知道到底那个车会不会掉进山谷里。。。。。

kensai

原帖由 新人王伏羲 于 25-4-2008 09:56 AM 发表

但是电子是如何和slit interfere 而产生interference pattern呢？是不是没有答案的．．．．．

我上面有讲

The conclusion of all this is that there is no experiment that can tell us what the electrons are doing at the slits that does not also destroy the interference pattern. Thisseems to imply that there is no answer to the question of what is goingon at the slits when we see the interference pattern.

至于那个相对论的问题，pole and barn paradox 我明白啦但是情况和我讲的问题有些不一样。所以还是不知道到底那个车会不会掉进山谷里。。。。。

如果以真实情况来讲，
我还是会以projectile motion 来看待这件事情，毕竟 Einstein 的posulate 是 fundamental law of physics apply in all inertial frames.


比如说，让我们来define ‘跌进山谷’
如果车在天空上的时间=T
Vertical distance = gT
如果gt> H(车的高度)   
车就会跌进山谷，这样没问题吧？

let H = 2 metre,
velocity of car = 0.9 c
gamma = 2.29
length of valley =3 x 10^7 km


Ok,现在如果在外面的科学家算到外面的 time 是不会跌下去，(gt <H)

time t= 0.2 seconds, 刚刚好 t= gh,就当他掉不下

但是他们担心因为proper time（ t') 在 车子的frame 比较长，而H 是constant,那么 gt' >H
这样车不就会掉下去？？？

t'= gamma x t = 0.458
gt'= 4.58 metre
高过车身 2 metre, 这样车肯定撞墙了

但是这不会发生，

因为在车手的frame 里，
length of valley' = 3 x 10^7 km / gamma = 1.31 x10^7 km
time floating in air = 1.31 x10^7 km / 0.9 c = 0.04 seconds,

g x t' = 0.3924 m
少过 H = 2metre
车还是不会掉下去


conclusion,
law of fundamental physics apply in all inertial reference frames,

还有我看到你的 effect 一定是等于 你看到我的effect,

不会说我看到你掉下去，但是你自己还在路上跑

hidden

In quantum theory, the state of a particle is describe by the wave function, \phi, which is a probabilistic function ( Prob = |\phi|^2 }. For time-dependent wave function, when you make an observation, the system is perturbed, and the wave function will collapse into one possible state. Observations change the system, because when a measurement is made, some external force must go into the system. For instance, if you use laser (which is most likely used in a quantum measurement) to probe the system, the momentum of the laser interacts with the particle and cause its wavefunction to collapse. However, for stationary (time-independent) systems, observations do not change the system, because the system is already in a &#160;stationary state

tensaix2j

原帖由 kensai 于 24-4-2008 01:51 AM 发表


每一粒电子都不知道而且不需要其他电子去了那里，
这个是independent event,

你玩骰子，每丢一次得到的号码都不知道前后得到的号码是多少，
但是你丢很多次很多次后，他们的机会率会自己平均成1/6,
但是骰 ...

请问这是不是 theory of large number ， 因为 数目越大， 1/6 就越明显。。。

但 我不明白的 是。。。
到底是怎么跟 机率 扯上关系的。。。 是不是 一个 matter 的 state 完全就是unpredictable 的？

antimatter

原帖由 hidden 于 28-4-2008 03:56 PM 发表
In quantum theory, the state of a particle is describe by the wave function, \phi, which is a probabilistic function ( Prob = |\phi|^2 }. For time-dependent wave function, when you make an observation ...

哦，原来是这样。那force 怎么让那个wave function collapse 呢？

hidden

原帖由 antimatter 于 28-4-2008 08:14 PM 发表 哦，原来是这样。那force 怎么让那个wave function collapse 呢？

对不起请容许我用英文，因为我学Physics都用英文，虽然我是华校生。Actually, I offered the physical interpretation of "wave function collapse". &#160; &#160;First, you need to bear in mind that whenever you see a thing, it is because the thing reflects light, that's why you can see it. It's the same when you are making a measurement. There are a lot of ways to measure things: radio waves, laser, X-rays etc. But no matter what&#160;method you use, you must interact with the system, in order to observe it. Now, in the quantum world, the energy of a particle is typically very small. So small that when it was hit by a photon (from the laser), its momentum changes, and hence its energy and other properties also have to change. Then, we say that the system is perturbed. When the system is perturbed, the wave function of the particle cannot be maintained, so it "collapses" in a sense. There is also mathematical equations to describe the phenomenon. If you know eigenfunctions, vector space, operator, you can get a better understanding of that. Hope that helps!&#160;
到底是怎么跟 机率 扯上关系的。。。 是不是 一个 matter 的 state 完全就是unpredictable 的？&#160;

I swear to you, this question is more interesting than you thought it would be. Did you know about "parallel universe" or "multiverse"? Maybe they will interest you! Why the quantum world is probabilistic has mystified a lot of people, including Einstein, so you're not alone. However, it is not entirely unpredictable. We can know certain properties of the system as long as it is not forbidden by Heisenberg's principle of uncertainty

[ 本帖最后由 hidden 于 28-4-2008 09:01 PM 编辑 ]

antimatter

collapse 是不是从wide range of momentum or energy collapse to one single momentum or energy 的意思？

kensai

原帖由 tensaix2j 于 28-4-2008 07:05 PM 发表



请问这是不是 theory of large number ， 因为 数目越大， 1/6 就越明显。。。

但 我不明白的 是。。。
到底是怎么跟 机率 扯上关系的。。。 是不是 一个 matter 的 state 完全就是unpredictable 的？

他是predictable 的，只是看准不准罢了，但是不可以determine specific location at specific time

[ 本帖最后由 kensai 于 29-4-2008 02:34 AM 编辑 ]

hidden

原帖由 antimatter 于 29-4-2008 12:12 AM 发表 collapse 是不是从wide range of momentum or energy collapse to one single momentum or energy 的意思？

可以这么说。不过，用wide range of possible states意思会更清楚。state (or eigenstate)在quantum里是很重要的concept...有兴趣的话，你应该去找关于Schrodinger's cat的相关资料……会学到很多新东西……

antimatter

哦，谢谢你。我还有问题……不知你愿意回答我吗？

就是怎么说那个double slit 的experiment 是time dependent 的？

你说wide range of possible states 这里是指eigen states 吗？possible states = eigen states? 据我所理解，eigenstates 是discrete 的，然后uncertainty principle 是不是continuous 的啊？如果是的话，那么那个system 是discrete 还是continuous 啊？比如momentum，在一个system 里面，你可以算出它的eigen value of momentum 是不是？也就是说momentum 只可以有几个discrete 的allowed state 是吧？可是uncertainty principle 又说他会有一定的range 是吧？那到底momentum 是必须discrete 还是要continuous 啊？

谢谢回答

hidden

你问到酱难，我差不多要当作回答考试题目来回答了 =P

就是怎么说那个double slit 的experiment 是time dependent 的？

不明白你的意思。time dependent 是用来形容particle的properties, 例如momentum, energy, position etc. 不明白一个experiment怎样time dependent...

你说wide range of possible states 这里是指eigen states 吗？possible states =eigen states?

a particle has discrete energy levels, right? So, given that a particle is at an energy level, its momentum can only be in certain states. These states are called momentum eigenstates. For example, an electron, if we know its energy level,then its momentum must be in one of these momentum eigenstates. However, in a more complex system (with 2 or more electrons/particles), the system can be in some linear combination of eigenstates. So if your system involves more than 2 particles, then possible states are not necessary equal to eigenstates. But possible states will still be some linear combinations of eigenstates.

据我所理解，eigenstates 是discrete 的，然后uncertainty principle是不是continuous 的啊？

eigenstates 的确是discrete 的。uncertainty principle是continuous 的？这是很奇怪的说法. Uncertainty principle states that {\Delta x} {\Delta p} >= h / 2, \Delta x = uncertainty in position, \Delta p = uncertainty in momentum, h = plack's constant. These uncertainties arise because of our measurements, not because of the system itself. In other words, uncertainty principle is applied to the measurements, not applied to the system or eigenstates.

那么那个system 是discrete 还是continuous啊？

Normally, the system itself is neither discrete nor continuous. Its properties are! like momentum, position, energy etc. Just like time dependency is referred to the properties of the system. See above.

比如momentum，在一个system 里面，你可以算出它的eigen value of momentum是不是？ 也就是说momentum 只可以有几个discrete 的allowed state 是吧？可是uncertaintyprinciple 又说他会有一定的range 是吧？那到底momentum 是必须discrete 还是要continuous 啊？

应该算被解释了吧？如果我解释得你不满意，我再想办法解释 anyway, 我觉得你有点confused了。通常我没听过一个system是discrete还是continuous的。如果有这种说法，normally应该是指一个system的energy。As energy level becomes higher, the difference between two energy levels become very small such that we can say the energy is almost continuous. In that case, the quantum theory is no longer used, but classical laws are used instead. So, to answer your question, at vey low energy states, the energy levels are discrete. At higher energy levels, the energy levels are very close together that they look like continuous. To see this, the energy level at 10th and 11th only differs by {1/10^2 - 1/11^2)*E, E is the gound state energy, which is typically of the order of 10eV. So the difference is of the order of 0.01 eV. and 1 eV is only roughly 10^-19 Joule. So you can safely say that in our world, the energy is continuous, unless, you discover how to measure the 10^-21 Joule difference in the energy! XD

[ 本帖最后由 hidden 于 30-4-2008 01:21 AM 编辑 ]

antimatter

Erm.. may be i din state my question in a proper manner.. let me ask again..

In the double slit experiment, when an observation (which slit the electron is going through) is made, the system is perturbed and collapsed to one possible state as you said in 51# right? This means that the wave function is time-dependent right? what is the factor that makes the wave function time-dependent?

For the second question, i think of two situations:
1. In the double slit experiment. The position (on the screen) where a single electron lands on, is determined by the electron's properties like momentum, energy, etc.. is it? However, these properties (momentum, energy, etc..) are discrete, am i correct? Hence, the position of the electron on the screen should be discrete also, rather than like a distribution of a probability right? But this is not match with what is happening.. we are considering a single electron in this case. so there is no linear combination of states in this case..

2. In an atom, (let say hydrogen atom for simplicity) the properties of the electrons (the energy, angular momentum, etc) are discrete right? Once we know the state of the atom, we have the 4 quantum number of the electron, then we will precisely know the energy, momentum, spin etc of the the electrons right? these properties are discrete right? however, the discrete properties still give rise to a distribution of the position of the electron. Meaning the electron doesn't stay in a circular motion with a constant radius r, indeed it is moving in a region where the probability is high. So, my question is that why the discrete properties give rise to a distribution of the position of the electron? Notice that this is happened within one atom, and not a linear combination of many atoms.

Sorry if i make you like answering an examination question..
Thank you so much for answering me

hidden

原帖由 antimatter 于 30-4-2008 11:49 AM 发表
This meansthat the wave function is time-dependent right? what is the factor thatmakes the wave function time-dependent?

Yes. The wavefunction collapses, so you can see its displacement (with some uncertainty of course!). This does not imply wavefunction is time dependent. The wavefunction is time dependent because its energy develops with time. (I think thats the definition) Time-independent wavefunction is usually used to simplify problems in textbook. You hardly see a time-independent system in nature (just like, there's never a 1-dimensional potential well). What factors cause the time dependency? I wish I knew . It can be derived mathematically, I see proofs like that, but I have not learned it yet (along the line that the Hamiltonian of a particle is a function of time, I think)

In the double slit experiment. The position (on the screen) where asingle electron lands on, is determined by the electron's propertieslike momentum, energy, etc.. is it? However, these properties(momentum, energy, etc..) are discrete, am i correct? Hence, the position of the electron on the screen should be discrete also, ratherthan like a distribution of a probability right? But this is not matchwith what is happening..

Everything you said is right here. It seems continuous exactly because the difference in possible displacements is so very small it's hard to detect on the large scale. (To get an electron to go forward, you accelerate it, so it is way above its ground state energy). So if you look at those famous pictures from electron double-slit experiment, your eyes cannot tell the discreteness within each band! Also, because the electron might hit at different places at the double slit, the outcome (of a single electron) is unpredicatable. But, if you gather enough electrons, we do see that the electron will experience interference much like any wave does, so we see the interference pattern.

Once we know the state of the atom, we have the 4 quantum number of theelectron, then we will precisely know the energy, momentum, spin etc ofthe the electrons right? these properties are discrete right? however,the discrete properties still give rise to a distribution of theposition of the electron. Meaning the electron doesn't stay in acircular motion with a constant radius r, indeed it is moving in aregion where the probability is high. So, my question is that why thediscrete properties give rise to a distribution of the position of theelectron? Notice that this is happened within one atom, and not alinear combination of many atoms.

If you know exactly the momentum of an electron, then the location of the electron has infinite uncertainty, i.e. it can be just right next to you or in other galaxy we dont know. This is given by the uncertainty principle. If you know the 4 quantum numbers, you'll know the energy and the momentum, because both of them are allowed to be determined simultaneously (there's a term for this which I cant recall right now). But momentum and displacement are not allowed. But, if your question is, why do we have electron orbits at all, like shown here: http://en.wikipedia.org/wiki/Electron_configuration. Then I think the answer is, those are not the location of the electron. Those are the momentum distribution of an electron, e.g. the distribution of a 1s electron is spherically distributed around the nucleus. The radius can go on to infinity, but the probability of finding a 1s electron is also infinitesimally small. Also, in the illustrations, those orbits have well-defined boundary, but in fact, they don't.

[ 本帖最后由 hidden 于 30-4-2008 12:35 PM 编辑 ]