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Chapter
Discrete Models
In some problems the states that mix in a way are few; then, the method of matrices lends itself to insightful (if qualitative) descriptions. The simplest model is \(2 \times 2\) : $$ H = \left( \begin{...
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Chapter
Fano Resonances
All the solvable models that we have seen in previous chapters lead to continuum spectra for unbound particles and discrete spectra for bound states; for example, the H atom gave us an infinity of discrete sta...
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Chapter
Quantum Transport and Quantum Pum**
The usual laws of the circuits (Ohm’s law, Kirchoff’s law, and so on) are valid in the macroscopic world. However, when the linear dimensions are less than an electron mean free path, which may be on the order...
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Chapter
Theoretical Physics and Mathematics
Following the method of Galileo, Theoretical Physics uses Mathematics as natural and essential language to describe reality. But even Galileo would probably be surprised by the degree of success of the mathema...
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Chapter
Dirac’s Delta
Let us start with the Heavyside (Oliver Heaviside (1850–1925) was probably the first to use the \(\delta \) before Dirac, and the work of George also implies the concept. Often the names are not historical...
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Chapter
Curvilinear Coordinates and Curved Spaces
Even in flat Euclidean space it may be useful to use curvilinear coordinates; for instance, in 3d problems having central symmetry, we obtain an important simplification when the line element
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Chapter
Postulate 2
\(D \equiv \frac{d}{dx}\) , and also \(\hat{x}\) ( multiplies by x), are examples of linear operators \(\hat{O}\) : \(\hat{O}(\varPhi +\varPsi )=\hat{O}\varPhi + \hat{O}\varPsi \) . Coordinates...
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Chapter
Postulate 4
The time evolution of \(\psi \) is governed by the Schrö equation $$\begin{aligned} i \hbar \frac{\partial \psi }{\partial t}= \hat{H}(t) \psi (t), \end{aligned}$$ where \(\hat{H}(t)\) is the Hamil...
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Chapter
Stationary States of One Particle in 3 Dimensions
The 3-dimensional plane wave is the product of one-dimensional plane waves and the kinetic energy is the sum of the contributions of motions along x, y, z. More generally, the problem is separable into Cartesian ...
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Chapter
Variational Principle for Schrödinger–Pauli Theory
The Schrödinger–Pauli theory is characterized by the fact that every system must have a ground state, whose energy is a lower bound to the energies of all states. This is not true in Classical Mechanics, in wh...
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Chapter
Pancharatnam Phase and Berry Phase
The Indian physicist S. Pancharatnam, quantum Optics in 1956, introduced the novel concept of a geometrical phase. Let \(H(\xi )\) denote a Hamiltonian that depends on some parameters \(\xi \) , with...
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Chapter
Entanglement, Its Challenges and Its Applications
When we dealt with the H atom in Chap. 17, we started from the classical formulation in terms of an effective particle having a reduced mass (see Sect. ...
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Chapter
Some Consequences of Maxwell’s Equations
The (classical) electromagnetic fields in vacuo that satisfy given boundary conditions can be calculated through Maxwell’s equations. In the Gauss system they read as:
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Chapter
The Eigenvalue Equation and the Evolution Operator
The operators that represent variables of classical dynamics are built by analogy with classical analogues. (Since the classical description, as we know, can be changed by canonical transformation, this statem...
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Chapter
The Postulates of Quantum Mechanics: Postulate 1
The system referred to above could be a particle, an atom, or even a macroscopic superconductor (then, x stands for a very large set of coordinates), so the statement is quite strong and general. In any case, all...
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Chapter
Postulate 3
Note that the system might have a large number of degrees of freedom, yet one can make a measurement involving one of them, like one component of angular momentum, which has an eigenvalue equation depending on...
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Chapter
The Quantum Harmonic Oscillator
The oscillator Hamiltonian in the coordinate representation is: $$\hat{H} = {p^{2} \over 2m} +\frac{1}{2} m \omega ^{2} x^{2}.$$ ...
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Chapter
Perturbation Theory
Suppose we can find the bound states of some Hamiltonian \(H_{0}\) by solving exactly the time-independent Schrödinge...
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Chapter
Spin and Magnetic Field
Classically, a point charge that circulates on a ring of radius r produces a current \(i = \frac{ev}{2 \pi r}\) , whic...
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Chapter
Thermal Physics
Thermodynamics is an axiomatic part of Theoretical Physics, which is presented as a set of phenomenological axioms or principles. Any investigation into the reasons why the principles are true and how they are...
