سؤالان عاجلان....

[محبة الرسول]

السلام عليكم ورحمة الله....

لدي سؤالان هامان بالنسبة الي...ارجو ان تتفضلوا بالاجابة....

can the state affect the observable

and if it can,how
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

light atoms are best treated by russell_saunders coupling and very heavy atoms by jj coupling,why
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

[deadheart]

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states. In quantum physics, on the other hand, the relation between system state and the value of an observable is more subtle, requiring some basic linear algebra for its de******ion. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions or state vectors.
In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state de******ion by a single vector may be destroyed, being replaced by a statistical ensemble. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this de******ion is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.
Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property. In the case of quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.
In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable. For example, suppose , with eigenvalue a, and exists in a d-dimensional Hilbert space, then
= a is made while the system of interest is in the state , then the eigenvalue a is returned with probability FPRIVATE "TYPE=PICT;ALT=\******style |\langle a|\phi\rangle|^2"(Born rule). One must note that the above definition is somewhat dependent upon our convention of choosing real numbers to represent real physical quantities. Indeed, just because dynamical variables are "real" and not "unreal" in the ****physical sense does not mean that they must correspond to real numbers in the mathematical sense. To be more precise, the dynamical variable/observable is a Hermitian operator in a finite-dimensional Hilbert Space and thus is represented by a Hermitian matrix. In an infinite-dimensional Hilbert space, the observable is represented by a symmetric operator, which may not be defined everywhere (i.e. its domain is not the whole space - there exist some states that are not in the domain of the operator). The reason for such a change is that in an infinite-dimensional Hilbert space, the operator becomes unbounded, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space, where every operator is bounded - it has a largest eigenvalue. For example, if we consider the position of a point particle moving along a line, this particle’s position variable can take on any number on the real-line, which is uncountably infinite. Since the eigenvalue of an observable represents a real physical quantity for that particular dynamical variable, then we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space, since the field we’re working over consists of the real-line. Nonetheless, whether we are working in an infinite-dimensional or finite-dimensional Hilbert space, the role of an observable in quantum mechanics is to assign real numbers to outcomes of particular measurements; this means that only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable.


ده جواب السؤال الاول وانا طلتو من بحث اذا كان هوا واذا منفعش حشوف

[محبة الرسول]

جزاك الله خيرا على المساعدة, ولكن ما اكثر فقرة تعتبر اجابة السؤال...اي اريد اختصارا...والموضوع يرتبط

بميكانيكا الكم وليس الكلاسيكية...فبعد اذنك لو تقدر تشوفلي اقرب اجابة..مشكور..

[رجب مصطفى]

بالنسبة للسؤال الثاني ...


In light atoms, the interactions between the orbital angular momenta of individual electrons is stronger than the spin-orbit coupling between the spin and orbital angular momenta. These cases are described by "L-S coupling". However, for heavier elements with larger nuclear charge, the spin-orbit interactions become as strong as the interactions between individual spins or orbital angular momenta. In those cases the spin and orbital angular momenta of individual electrons tend to couple to form individual electron angular momenta


وهذه مقارنة بين النوعين !!!


Russell –Saunders Coupling
Assumes no coupling between individual electron’s angular and spin momenta
J is from sum of all L and S
Work’s for light atoms

jj – coupling
Spin and orbital momentum of each electron is coupled strongly
Must treat electron as particle with angular momentum j
S and L are not ‘true’ quantum numbers, only J
Selection rules for DS can be broken in heavy atoms because only DJ selection rule is valid

[رجب مصطفى]

وهذه إضافة أخرى أكثر تفصيلاً ... من ويكيبديا ...


LS coupling

In light atoms (generally Z < 30), electron spins si interact among themselves so they combine to form a total spin angular momentum S. The same happens with orbital angular momenta li, forming a total orbital angular momentum L. The interaction between the quantum numbers L and S is called Russell–Saunders coupling or LS coupling. Then S and L add together and form a total angular momentum J

This is an approximation which is good as long as any external magnetic fields are weak. In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the Paschen–Back effect.), and the size of LS coupling term becomes small

For an extensive example on how LS-coupling is practically applied, see the article on term symbols

JJ coupling

In heavier atoms the situation is different. In atoms with bigger nuclear charges, spin-orbit interactions are frequently as large or larger than spin-spin interactions or orbit-orbit interactions. In this situation, each orbital angular momentum li tends to combine with each individual spin angular momentum si, originating individual total angular momenta ji. These then add up to form the total angular momentum J

This de s c r i b t i on, facilitating calculation of this kind of interaction, is known as jj coupling


http://en.wikipedia.org/wiki/Angular_momentum_coupling (http://en.wikipedia.org/wiki/Angular_momentum_coupling)

[deadheart]

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator’s spectrum is discrete, the observable can only attain those discrete eigenvalues.
The time evolution of a quantum state is described by the Schr&ouml;dinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

وده ينفع ولا لا

[deadheart]

شكراً استاذ رجب

واي حاجة تاني رد وانا اشوفو على طول

[محبة الرسول]

جزاكما الله كل خير....ولكن هل من توضيح اكثر في السؤال الاول..؟؟

معلش تعبتكما....بس السؤال ده المفروض انه مطروح من قبل الدكتور...وعلينا البحث عن اجابته لأنه

سيأتي في الامتحان وانا بحاجة الى فهمه مع الايجاز لأستطيع الاجابة عنه...

[deadheart]

شوفي ده على وجه السرعة

هنا (http://en.wikipedia.org/wiki/Quantum_mechanics)

[رجب مصطفى]

بالنسبة للسؤال الثاني ... فإجابتي عليه كانت سريعة لأني كنت أذاكر هذا الموضوع في نفس الوقت !!!

أما الأول ... فأنا بصراحة لم أفهم المطلوب منه سابقاً ... كما تعلمين ... لذا لا أستطيع أن أعطيكِ إجابة شافية وافية له !!!

وصلي اللهم على المصطفى "محمد بن عبد الله" وعلى آله الأطهار وصحبه الأخيار أجمعين ...

[محبة الرسول]

جزاكما الله خيرا وبارك فيكما....لا تنسيا الصلاة على الحبيب المصطفى

[deadheart]

لا شكر على واجب

اللهم صلي وسلم على اشرف الخلق والمرسلين سيدنا محمد النبي الامين

[محبة الرسول]

عذرا استاذ موسى بس لو ممكن تشرحلي الفقرة ديه عشان انا مش مستوعباها كويس, وحاسة انها الاجابة...

In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space (http://en.wikipedia.org/wiki/Hilbert_space), the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state de******ion by a single vector may be destroyed, being replaced by a statistical ensemble (http://en.wikipedia.org/wiki/Statistical_ensemble).

[deadheart]

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