For greater resolution, simply click the links for "medium" or "large" in the frame of any given page. Scattering Problems and Born Approximation. Examples of Quantized Field Theories.

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Theory of Line Shift and Line Width. Free Particle Scattering Problems. It was the central topic in the famous Bohrâ€”Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied.

Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled , so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a probability distribution ; that is, it describes the probability of obtaining the possible outcomes from measuring an observable.

Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate but better than the Bohr model whereby electron location is given by a probability function , the wave function eigenvalue , such that the probability is the squared modulus of the complex amplitude , or quantum state nuclear attraction. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable.

These are known as eigenstates of the observable "eigen" can be translated from German as meaning "inherent" or "characteristic". In the everyday world, it is natural and intuitive to think of everything every observable as being in an eigenstate.

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Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum since they are conjugate pairs or its energy and time since they too are conjugate pairs. Rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values eigenstates.

Usually, a system will not be in an eigenstate of the observable particle we are interested in. However, if one measures the observable, the wave function will instantaneously be an eigenstate or "generalized" eigenstate of that observable. This process is known as wave function collapse , a controversial and much-debated process [33] that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wave function collapsing into each of the possible eigenstates.

## Advanced Quantum Mechanics - / - University of Cologne

For example, the free particle in the previous example will usually have a wave function that is a wave packet centered around some mean position x 0 neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result. After the measurement is performed, having obtained some result x , the wave function collapses into a position eigenstate centered at x.

During a measurement , on the other hand, the change of the initial wave function into another, later wave function is not deterministic, it is unpredictable i. A time-evolution simulation can be seen here. Wave functions change as time progresses. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain with time.

This also has the effect of turning a position eigenstate which can be thought of as an infinitely sharp wave packet into a broadened wave packet that no longer represents a definite, certain position eigenstate. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static, spherically symmetric wave function surrounding the nucleus Fig.

Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the "wave-like" behavior of quantum states. There exist several techniques for generating approximate solutions, however. In the important method known as perturbation theory , one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the simpler model by for one example the addition of a weak potential energy.

Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos. There are numerous mathematically equivalent formulations of quantum mechanics.

Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in for the creation of quantum mechanics, the role of Max Born in the development of QM was overlooked until the Nobel award. The role is noted in a biography of Born, which recounts his role in the matrix formulation of quantum mechanics, and the use of probability amplitudes. Heisenberg himself acknowledges having learned matrices from Born, as published in a festschrift honoring Max Planck. Examples of observables include energy , position , momentum , and angular momentum.

Observables can be either continuous e. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The rules of quantum mechanics are fundamental. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle , which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or, equivalently, larger quantum numbers, i.

In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an established classical model of a particular system, then attempt to guess the underlying quantum model that would give rise to the classical model in the correspondence limit.

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory , which applies quantization to a field rather than a fixed set of particles.

The first complete quantum field theory, quantum electrodynamics , provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field.

This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics , and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory , by the physicists Abdus Salam , Sheldon Glashow and Steven Weinberg.

These three men shared the Nobel Prize in Physics in for this work.

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## 124.325 Advanced Quantum Physics (15 credits)

It has proven difficult to construct quantum models of gravity , the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity the most accurate theory of gravity currently known and some of the fundamental assumptions of quantum theory.

The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity. Classical mechanics has also been extended into the complex domain , with complex classical mechanics exhibiting behaviors similar to quantum mechanics. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy. For microscopic bodies, the extension of the system is much smaller than the coherence length , which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.

A big difference between classical and quantum mechanics is that they use very different kinematic descriptions. In Niels Bohr 's mature view, quantum mechanical phenomena are required to be experiments, with complete descriptions of all the devices for the system, preparative, intermediary, and finally measuring. The descriptions are in macroscopic terms, expressed in ordinary language, supplemented with the concepts of classical mechanics.

Quantum mechanics does not admit a completely precise description, in terms of both position and momentum, of an initial condition or "state" in the classical sense of the word that would support a precisely deterministic and causal prediction of a final condition. For a stationary process, the initial and final condition are the same. For a transition, they are different. Obviously by definition, if only the initial condition is given, the process is not determined.

For many experiments, it is possible to think of the initial and final conditions of the system as being a particle.

In some cases it appears that there are potentially several spatially distinct pathways or trajectories by which a particle might pass from initial to final condition. It is an important feature of the quantum kinematic description that it does not permit a unique definite statement of which of those pathways is actually followed. Only the initial and final conditions are definite, and, as stated in the foregoing paragraph, they are defined only as precisely as allowed by the configuration space description or its equivalent.

In every case for which a quantum kinematic description is needed, there is always a compelling reason for this restriction of kinematic precision. An example of such a reason is that for a particle to be experimentally found in a definite position, it must be held motionless; for it to be experimentally found to have a definite momentum, it must have free motion; these two are logically incompatible. Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces.

This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage. Hamiltonian dynamics can be used for this. Classical kinematics also allows the description of a process analogous to the initial and final condition description used by quantum mechanics.

Lagrangian mechanics applies to this. Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence , and while they do not directly contradict each other theoretically at least with regard to their primary claims , they have proven extremely difficult to incorporate into one consistent, cohesive model.

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Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant " Theory of Everything " TOE.

Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics.