the stern-gerlach experiment 1. the stern-gerlach experiment (sge) is performed in 1921, to see if...
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The Stern-Gerlach Experiment
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• The Stern-Gerlach Experiment (SGE) is performed in 1921, to see if electron has an intrinsic magnetic moment.
• A beam of hot (neutral) Silver (47Ag) atoms was used.• The beam is passed through an inhomogeneous magnetic field
along z axis. This field would interact with the magnetic dipole moment of the atom, if any, and deflect it.
• Finally, the beam strikes a photographic plate to measure,if any, deflection.
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• Why Neutral Silver atom? – No Lorentz force (F = qv x B) acts on a neutral atom,
since the total charge (q) of the atom is zero.– Only the magnetic moment of the atom interacts with
the external magnetic field.– Electronic configuration: 1s2 2s2 2p6 3s2 3p6 3d10 4s1 4p6 4d10 5s1
So, a neutral Ag atom has zero total orbital momentum.
– Therefore, if the electron at 5s orbital has a magnetic moment, one can measure it.
• Why inhomogenous magnetic Field?– In a homogeneous field, each magnetic moment
experience only a torque and no deflecting force.– An inhomogeneous field produces a deflecting force
on any magnetic moments that are present in the beam.3
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Spin• we are going to try to understand the spin of the electron. First we
need to review the interaction of magnetic moments with a magnetic field
• The electron happens to have a magnetic moment which we will assume comes from the spin. We are going to try to understand what causes this. For the moment we will picture a magnetic moment as a loop of current I with a side L and area A like so:
Now A=L2 and the magnetic moment=IA. There is also a magnetic field B pointing out of the page. The potential energy of the loop is
A I
B
so if we have an electron with a spin s which we will assume is proportional
to , (the value of the proportionality constant will have to be measured in
experiment) i.e. s we will ha
U B
ve for the electron that its potential
energy in a magnetic field is also U B 4
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We want to figure out some way to figure out which way the spin is pointing – so if we could make a force which depends on the direction of spin – this would do it.
z
We know that force is a derivative of the potential energy.
So let's work in the z direction
F ( ) where I have assumed
is a constant and that B changes only in the z direct
Zz
BUB
z z z
ion
(I will set up the experiment that way) So this says that if
I have a changing B field that there will be a force on the
electron depending on the z component of or in other
words, depending on
the direction of the spin with respect
to the magnetic field 5
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• So can we make a magnetic field which has a changing z component in the z direction i.e. a dBz/dz? We can. We shape our pole tips as follows where the field lines are in red. We remember that the strength of the B field is proportional to the density of lines. The density is highest near the bottom pointed tip so the field gets stronger toward the bottom and dBz/dz points downward
higher B
lower B
dBz/dz
Now since zF Zz
B
z
and and s point in opposite directions (the electron is negatively charged) thenif z> 0 (sz<0) so Fz is if z< 0 (sz>0) so Fz is
N
S
Basically the force on the electron isproportional to the component of the spin in the z direction
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• In the real experiment, an electron beam was not used. It was a Silver beam. Silver has one outer electron with an angular momentum of L=0. So the entire electron’s magnetic moment is due to the spin of the outer electron. But for us, lets think of electrons. We set the experiment up as follows, and then look to see what pattern we see on the screen
dBz/dz
screen
Beam of “electrons”N
S
7
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what might we expect? Classically you might expect one wide spread reflecting the fact that the spin is randomly pointed and hence the z component of the magnetic momentum is spread around. Note that this initially random direction in like a ball in 4
what do we really see?
electrons
N
S
zB
z
spin direction
+z In reality, the beam is split neatly in half, as if the beam was initially ONLY pointed either in the +z or –z direction.
electrons
N
S
zB
z
spin direction
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spin direction+z
z
Now I can make a similar Stern-Gerlach aparatus which picks out the x component
or y component of , by changing different components of the B field as follows.
F ( ) or xx y
BUB
z z z
These have to be cleverly designed but I will not worry about that.
yB
z
So remember SGx will put a force in the z direction on the component ofthe spin in the x direction. Similarly for y and z
N
S
, or zx yB
z
SGx, y or z
electrons
I will represent the apparatus as a box so we can have a box which is labeled SGx, SGy, and SGz which push on the various components of
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N
S
SGx
electrons
Now lets try putting it through an SGx apparatus
+z
y
x
N
S
SGz
SGz+
SGz-Block
N
S
SGz
Now lets try using two SG apparatus. We use an SGz and then block off the bottom“-” component before we let it go through the second. Not surprisingly only the top bump is present. There is no SGz- compontent. It looks like once its “+z” it always stays “+z”
SGz+
we see that it still spits in two as if the spin were either in the +x or –x direction !
Maybe somehow the electrons “guess” which direction the apparatus is set and align themselves accordingly
SGx+
SGX-
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N
S
SGz
SGz+
SGz-Block
N
S
SGx
Now supposing we do the same thing, but this time, instead of “testing” the second beam with SGz lets test it with SGx.
N
S
SGz
SGz+
SGz-Block
N
S
SGx
The +z beam, when passed through as SGX filter, gives both a +x and -x components, but note that the intensity is less! Its as if the SGz+ beam was made up of a combination of a +x and –x components
SGx+
SGx-
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N
S
SGz
SGz+
SGz-
Block
N
S
SGx
How about it we try the same with a –z?
N
S
SGz
SGz+
SGz-
Block
N
S
SGx
The -z beam, when passed through an SGX filter, also gives both a +x and -x components!
SGx+
SGx-12
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N
S
SGz
SGz+
SGz-Block
N
S
SGx
Block
N
S
SGz
Now lets try using SGx to split up a SGz+ beam up into + and – x components and then test this to see if there is a –z component. Note this is exactly like the situation before where we used two SGz filters but we have added and extra SGx in between.
We seem to get the –z component back again. Its like theSGx filter has regenerated the SGz- back.
SGx-
SGx+
N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
N
S
SGz
SGx+
SGx-
SGz+
SGz-
This rather odd situation where a component of the beam is regenerated, is reminiscent of polarized light where we areable to make a set of two perpendicular polarizers passsome light through by adding a third on at 45 in between
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N
S
SGz
SGz+
SGz-Block
N
S
SGy
Now we remember there was also a SGy polarizer. Lets use it.
When we pass an SGz+ beam thorough an SGy filter we again see two components.
SGy+
SGy-
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electrons
N
S
zB
z
spin direction
+z ;zs
;zs
N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
N
S
SGz
SGx+
SGx-
SGz+
SGz-
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N
S
SGx
electrons
Now lets try putting it through an SGx apparatus
+z
y
x
SGx+
SGX-
N
S
SGz
N
S
SGz
SGz+
SGz-Block
SGz+
SGz+
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N
S
SGz
SGz+
SGz-
N
S
SGy
electrons
N
S
B
z
spin direction
+z
Block
SG?
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N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
N
S
SGz
SGx+
SGx-
SGz+
SGz-
Block
A
|ai
Block
B
Block
C
|bj
|ck
Block
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A
|ai
Block
C
|ck
Block
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N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
N
SSGz
SGx+
SGx-
SGz+
SGz-
This rather odd situation where a component of the beam is regenerated, is reminiscent of polarized light where we areable to make a set of two perpendicular polarizers passsome light through by adding a third on at 45 in between
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نتیجه؟
باید الکترون، اسپین حالت بررسی برای نیاز مورد بعدی دو برداری فضای.مختلط باشد
عملگرها و براها کتها،
کت فضایماهیت به بسته آن ابعاد که است مختلط برداری فضای یک کت فضای
. میشود انتخاب بررسی، مورد سیستم فیزیکی
گرالخ – - اشترن دستگاه ( آزمایش از عبور هنگام دوبعدی اسپین( است ممکن مسیر دو فقط
پیوسته - ممکن طیفهای حاالت تعداد ذره تکانه یا مکانشمارش غیرقابل
نامتناهی برداری فضای هیلبرت فضای
یک* با فیزیکی حالت حالت هر داده بردار نشان مختلط برداری فضای دررا. برداری چنین حالت میشود گویند کت
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فیزیکی حالت اطالعات کلیه حاوی حالت کت. است بررسی تحت
میدهند؟ نشان را یکسان فیزیکی حالت یک و آیا
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پذیر مشاهده عملگر Observableهر یک شکل به اسپین، و ممانتوم مانند ،Operator مانند ، A ،
. میشود داده نمایش برداریش فضای در
. میکند اثر حالت کت به چپ از همواره عملگر
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نامیده حالت ویژه کت، ویژه یک با متناظر فیزیکی .eigenstateحالت میشود
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فضای کلی حالت نوعی Nدر عملگر بردارهای ویژه از متشکل :Aبعدی نوشت زیر شکل به میتوان را
برا فضای
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داخلی ضرب
. میگیرند قرار استفاده مورد کت فضای از بردار یک و برا فضای از بردار یک داخلی، حاصلضرب .برای میگیرند قرار استفاده مورد کت فضای از بردار یک و برا فضای از بردار یک داخلی، حاصلضرب برای
داخلی ضرب اساسی دوخاصیت
1: همند- مختلط مزدوج و
مثبت- 2 معین مقدار33
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: متعامد کتهای
: نرمالیزه کتهای
عملگرها:
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: برابرند عملگر دو اگر
:( ) زمانی تحول عملگر از غیر پذیرند شرکت و جابجاپذیر عملگرها جمع
: هستند خطی درس این در استفاده مورد عملگرهای تمام
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: داریم بلکه نیستند، هم همزاد کلی حالت در برا و کت
: هرمیتی عملگر
: عملگرها ضرب
: است- پذیر شرکت ولی نیست پذیر جابجا
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: ضرب انواع
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: پذیری شرکت قضیه
1
2
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باشد Xاگر هرمیتی عملگر
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قضیه:هرمیتی عملگر یک مقادیر آن، حقیقی ویژه متفاوت مقادیر ویژه با متناظر کتهای ویژه و ند.متعامداند
: نوشت میتوان است، هرمیتی عملگر چون
( )
( )
: و یا هستند : یکی نیستند 40 ویا
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. در دلخواه کت هر لذا میدهند را اورتونرمال کامل مجموعه یک تشکیل هرمیتی عملگر نرمالیزه کتهای ویژه که دیدیم: داد بسط آنها برحسب میتوان را کت برداری فضای
بودن کامل رابطه41
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: داریم باشد نرماالیز اگر رابطه در پس
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: داریم میدهیم، اثر روی را عملگر
. . گویند تصویر عملگر عملگر، این به لذا میدهد را پایه کت روی بر تصویر حقیقت در که
بودن کامل رابطه
43
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: داریم هرمیتی عملگر یک برای
44
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: عملگرها ضرب
ضرب ماتریسی نمایش
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46
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عملگر کتهای ویژه .Aاگر است قطری ماتریس باشند، برداری فضای های پایه ،
47
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Szعملگر Szعملگر
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: میکنیم معرفی هستند، هرمیتی غیر که را زیر عملگر دو
?
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گیری اندازه؟
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A careful analysis of the process of observation in atomic physics has shown that the subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between the preparation of an experiment and the subsequent measurement.
- Erwin Schrödinger (1887-1961)
گیری اندازه
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گیری Measurementsاندازه
It is impossible to determine into which eigenstate a given system will jump, but it ispossible to predict the probability of such a transition.
What is the probability that a system in some initial state makes a transition to an
eigenstate of an observable , as a result of a measurement made on the system?
Let us start with the simplest case. If the system is initially in an eigenstate then the
transition probability to a eigenstate corresponding to a different eigenvalue is
zero, and the transition probability to the same eigenstate is unity. It is convenient
to normalize our eigenkets such that they all have unit norms. It follows from the
orthogonality property of the eigenkets that
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Can we use this correspondence to obtain a general rule for calculating transition probabilities(value of the inner product)?
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پذیر مشاهده روی گیری اندازه از قبل کنیم : Aفرض شود داده نشان زیر خطی ترکیب شکل به سیستم ،
electrons
N
S
zB
z
spin direction
+z
: باشد داشته قرار پذیر مشاهده حالت ویژه یک در سیستم اینکه مگر دهد می تغییر را حالت معموال گیری اندازه
54
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55
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مجموعه روی بر گیری اندازه زیادی تعداد باید عمال احتمال تعیین برای میکنیم، صحبت تنها سیستم یک از ما اینکه وجود با
. دهیم صورت هستند، یکسان حالت کت دارای که یکسان فیزیکی سیستمهای از ای
را ای مجموعه .Pure ensemble چنین گویند
در سیستم یافتن احتمال ، باشد، داشته قرار پذیر مشاهده کت ویژه یک در ابتدا سیستم اگر فوق، رابطه به توجه با
. است یک برابر
( ) . زمانی: تحول گیرد انجام پی در پی صورت به فوق گیریهای اندازه باید حتما توجه
. است صفر مثل دیگر حالت ویژه یک در سیستم یافتن احتمال همینطور
* باشد* منفی نباید احتمال 56
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Expectation value
Expectation value & eigenvalues?
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58
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N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
N
S
SGz
SGx+
SGx-
SGz+
SGz-
0.5احتمال
حقیقی
59
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تعامد
:عملگرهرمیتی
نوشت میتوان ترتیب همین :به
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کرد؟ تعیین را فازها میتوان چگونه
N
S
SGz
SGz+
SGz- Block
N
S
SGx
Block
SGx+
SGx-N
S
SGy
61
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ماتریسی: های مولفه همه .Syو Sxنتیجه باشند حقیقی نمیتوانند . است موهومی حتما دیگری است، حقیقی یکی های مولفه اگر
?
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است: اهمیت دارای نسبی فاز توجه
تعریف:
: جابجایی رابطه
پاولی طرد 63!!اصل
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:تعریف
64
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Compatible Observables
چیست؟ سازگار پذیر مشاهده دو کتهای ویژه بین رابطه
Degeneracyتبهگنی؟
کتهای Bو Aاگر ویژه با هم میتوان را مشخص کت فضای یک باشند، کتهای Aسازگار ویژه با هم .Bو داد پوشش
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ماتریسی های :مولفه
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عملگر Bعملگر کتهای ویژه های پایه ویژه Aدر این از خطی ترکیب میتوان ولی نیست قطری
که کرد پیدا کتها
. کنند قطری آنرا
نتیجه؟
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68
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!! هایزنبرگ قطعیت عدم اصل
. نیستند همزمان کتهای ویژه داری ناسازگار پذیرهای مشاهده
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70
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. ها از عبور در از آوردن بدست احتمال
71
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72
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The Heisenberg Uncertainty Principle
نوشت میتوان حالتی هر برای کنید :ثابت
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75
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اثبات:
باال رابطه راست :سمت
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77
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. شود داده پوشش دیگری یا یکی کتهای ویژه با میتواند نظر مورد کت فضای ناسازگار، پذیر مشاهده دو برای
( چیست؟ ( نمایش توصیف دو این بین ارتباط
: پایه کتهای مجموعه تغییر
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79
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: بعد سه در دوران ماتریس شبیه
چیست؟ دیگر پایه در بسط ضریب پایه، یک در بسط ضریب بودن معلوم با
های پایه در ستونی ماتریس به مربعی ماتریس اعمال با فقط جدید های پایه در کت یک برای ستونی ماتریس یعنی: شود حاصی قدیمی
80
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اسپین: 1مثال
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82
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اسپین: 2مثال
83
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: جدید و قدیم ماتریسی های مولفه بین رابطه
: نمایش از مستقل
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تمرین
85
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86
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87
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88
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Diagonalization
کرد پیدا را هایی پایه میتوان چگونه بدانیم، عملگر کتهای ویژه از غیر هایی پایه در را عملگر یک ماتریسی های مولفه اگر
باشد؟ قطری درآن که
89
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، ویژه مقادیر همان .bjها هستند ،
90
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مثال:
91
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92
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کنید: قطری را روبرو ماتریس مثال
93
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Unitary Equivalent Observables
زمانی را پذیر مشاهده مقادیر Unitary Equivalent Observablesدو مجموعه دارای که گویند. باشند یکسان ویژه
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مثال:
95
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Continuous spectra
96