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**1. Quantum theory: introduction and principles**

1.1 Wave-particle duality 1.2 The Schrödinger equation 1.3 The Born interpretation of the wavefunction 1.4 Operators and theorems of the quantum theory 1.5 The Uncertainty Principle = c

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**1.1 Wave-particle duality **

A. The particle character of electromagnetic radiation The photoelectric effect e- (Ek) h metal The photon h can be seen as a particle-like projectile having enough energy to collide and eject an electron from the metal. The conservation of energy requires that the kinetic energy of the ejected electron should obey: ½mv2 = h - , called the metal workfunction, is the minimum energy required to remove an electron from the metal to the infinity. The ejection threshold of electrons does not depend on the intensity of the incident radiation.

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**B. The wave character of the particles**

Electron diffraction Diffraction is a characteristic property of waves. With X-ray, Bragg showed that a constructive interference occurs when =2d sin. Davidsson and Germer showed also interference phenomenon but with electrons! Particles are characterized by a wavefunction d V A link between the particle (p=mv) and the wave () natures An appropriate potential difference creates electrons that can diffract with the lattice of nickel

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**Example 1: Northern light**

The sun has a number of holes in its corona from which high energy particles (e-, p+, n0) stream out with enormous velocity. These particles are thrown out through our solar system, and the phenomenon is called solar wind. A part of this solar wind meets the earth’s magneto sphere, the solar wind particles are accelerated down to the earth along the open magnetic field lines. The field lines are open only in the polar regions. At lower latitudes the field is locked. That’s why we have the Northern Lights only in the polar regions. When the solar wind particles collide with the air molecules (O2, N2), their energy is transferred to excitation energy of the molecules. The excited molecules come back in their ground state by emitting light at specific frequencies: green-blue color from N2, red and green from O2. It is billions of such processes occurring simultaneously that produces the Northern Lights. Magnetic field of the earth

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**1.2 The Schrödinger Equation**

From the wave-particle duality, the concepts of classical physics (CP) have to be abandoned to describe microscopic systems. The dynamics of microscopic systems will be described in a new theory: the quantum theory (QT). A wave, called wavefunction (r,t), is associated to each object. The well-defined trajectory of an object in CP (the location, r, and momenta, p = m.v, are precisely known at each instant t) is replaced by (r,t) indicating that the particle is distributed through space like a wave. In QT, the location, r, and momenta, p, are not precisely known at each instant t (see Uncertainty Principle). In CP, all modes of motions (rot, trans, vib) can have any given energy by controlling the applied forces. In the QT, all modes of motion cannot have any given energy, but can only be excited at specified energy levels (see quantization of energy). The Planck constant h can be a criterion to know if a problem has to be addressed in CP or in QT. h can be seen has a “quantum of an action” that has the dimension of ML2T-1 (E= h where E is in ML2T-2 and is in T-1). With the specific parameters of a problem, we built a quantity having the dimension of an action (ML2T-1). If this quantity has the order of magnitude of h (~10-34 Js), the problem has to be treated within the QT.

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** In CP, the dynamics of objects is described by Newton’s laws**

In CP, the dynamics of objects is described by Newton’s laws. Hamilton developed a more general formalism expressing those laws. For a conservative system, the dynamics is described by the Hamilton equations and the total energy E corresponds to the Hamiltonian function H=T+V. T is the kinetic energy and V is the potential energy. This formalism appears to be close to that in which the dynamics of quantum systems is developed. Because of this similarity, the correspondence principles are proposed to pass from the CP to the QT: Classical mechanics Quantum mechanics Schrödinger Equation

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**(operator)(function)=(constant)×(same function) **

The Schrödinger Equation (SE) shows that the operator H and iħ/t give the same results when they act on the wavefunction. Both are equivalent operators corresponding to the total energy E. In the case of stationary systems, the potential V(x,y,z) is time independent. The wavefunction can be written as a stationary wave: (x,y,z,t)= (x,y,z) e-it (with E=ħ). This solution of the SE leads to a density of probability |(x,y,z,t)|2= |(x,y,z)|2, which is independent of time. The Time Independent Schrödinger Equation is: NB: In the following, we only envisage the time independent version of the SE. or The Schrödinger equation is an eigenvalue equation, which has the typical form: (operator)(function)=(constant)×(same function) The eigenvalue is the energy E. The set of eigenvalues are the only values that the energy can have (quantization). The eigenfunctions of the Hamiltonian operator H are the wavefunctions of the system. To each eigenvalue corresponds a set of eigenfunctions. Among those, only the eigenfunctions that fulfill specific conditions have a physical meaning.

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**1.3 The Born interpretation of the wavefunction**

Example of a 1-dimensional system Physical meaning of the wavefunction: If the wavefunction of a particle has the value (r) at some point r of the space, the probability of finding the particle in an infinitesimal volume d=dxdydz at that point is proportional to |(r)|2d |(r)|2 = (r)*(r) is a probability density. It is always positive! Hence, if the wavefunction has a negative or complex value, it does not mean that it has no physical meaning… because what is important is the value of |(r)|2 ≥ 0; for all r. But, the change in sign of (r) in space (presence of a node) is interesting to observe in chemistry: antibonding orbital overlap (see chap 4: Electronic structure in molecules). Node

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**A. Normalization Condition**

The solution of the differential equation of Schrödinger is defined within a constant N. Indeed, if ’ is a known solution of H’=E’, then =N’ is a also solution for the same E. H=E ⇔ H(N’)= E(N’) ⇔ N(H’)=N(E’) ⇔ H’=E’ The mathematical expression of the eigenfunction should be such that the sum of the probability of finding the particle over all infinitesimal volumes d of the space is 1. That insures the particle to be present in the space: Normalization condition. We have to determine the constant N, such that the solution =N’of the SE is normalized. B. Other mathematical conditions (r)≠∞ ; ⍱r → if not: no physical meaning for the normalization condition (r) should be single-valued ⍱r → if not: 2 probability for the same point!! The SE is a second-order differential equation: (r) and d(r)/dr should be continuous

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**C. The kinetic energy and the wavefunction**

A particle is expected to have a high kinetic energy if the average curvature of its wavefunction is high. The kinetic energy is then a kind of average over the curvature of the wavefunction: a large contribution to the observed value originates from the regions where the wavefunction is sharply curved ( 2 / x2 is large) and the wavefunction itself is large (* is large too).

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**Example 2: the wave function in a periodic system: electrons in a metal**

Schrödinger: periodic potential: Bloch theorem: Real part of the wavefunction for valence electrons in the potential created by the nuclei periodic

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**Example 3: Quantum corral created and observed with Scanning Tunneling Microscopy (STM)**

Scientists discovered a new method for confining electrons to artificial structures at the nanometer lengthscale. Surface state electrons on Cu(111) were confined to closed structures (corrals) defined by barriers built from Fe adatoms. The barriers were assembled by individually positioning Fe adatoms using the tip of a low temperature scanning tunneling microscope (STM). A circular corral of radius 71.3 Angstroms was constructed in this way out of 48 Fe adatoms. This STM image shows the direct observation of standing-wave patterns in the local density of states of the Cu(111) surface. These spatial oscillations are quantum-mechanical interference patterns caused by scattering of the two-dimensional electron gas off the Fe adatoms and point defects.

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**1.4 Operators and principles of quantum mechanics**

A. Operators in the quantum theory (QT) An eigenvalue equation, f = f, can be associated to each operator . In the QT, the operators are linear and hermitian. Linearity: is linear if: (c f)= c f (c=constant) and (f+)= f+ NB: “c” can be defined to fulfill the normalization condition Hermiticity: A linear operator is hermitian if: where f and are finite, uniform, continuous and the integral for the normalization converge. The eigenvalues of an hermitian operator are real numbers (= *) When the operator of an eigenvalue equation is hermitian, 2 eigenfunctions (fj, fk) corresponding to 2 different eigenvalues (j, k) are orthogonal.

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**B. Principles of Quantum mechanics**

1. To each observable or measurable property <> of the system corresponds a linear and hermitian operator , such that the only measurable values of this observable are the eigenvalues j of the corresponding operator. f = f 2. Each hermitian operator representing a physical property is “complete”. Def: An operator is “complete” if any function (finite, uniform and continuous) (x,y,z) can be developed as a series of eigenfunctions fj of this operator. 3. If (x,y,z) is a solution of the Schrödinger equation for a particle, and if we want to measure the value of the observable related to the complete and hermitian operator (that is not the Hamiltonian), then the probability to measure the eigenvalue k is equal to the square of the modulus of fk’s coefficient, that is |Ck|2, for an othornomal set of eigenfunctions {fj}. Def: The eigenfunctions are orthonormal if NB: In this case:

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4. The average value of a large number of observations is given by the expectation value <> of the operator corresponding to the observable of interest. The expectation value of an operator is defined as: 5. If the wavefunction =f1 is the eigenfunction of the operator (f = f), then the expectation value of is the eigenvalue 1. For normalized wavefunction 6. Two operators having the same eigenfunctions are “commutable”. Reciprocally, if two operators commute, they have a common “complete” set of eigenfunctions. Def: If the product of two operators is commutative, 12- 21= (12- 21)=0, then the operators are commutable. In this case, the commutator (12- 21), also written [1, 2], is equal to zero.

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**1.5 The Uncertainty Principle**

1. When two operators are commutable (and with the Hamiltonian operator), their eigenfunctions are common and the corresponding observables can be determined simultaneously and accurately. 2. Reciprocally, if two operators do not commute, the corresponding observable cannot be determined simultaneously and accurately. If (12- 21) = c, where “c” is a constant, then an uncertainty relation takes place for the measurement of these two observables: where Uncertainty Principle

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**Example 4: the Uncertainty Principle**

1. For a free atom and without taking into account the spin-orbit coupling, the angular orbital moment L2 and the total spin S2 commute with the Hamiltonian H. Hence, an exact value of the eigenvalues L of L2 and S of S2 can be measured simultaneously. L and S are good quantum numbers to characterize the wavefunction of a free atom see Chap 3 “Atomic structure and atomic spectra”. 2. Position x and momentum px (along the x axis). According to the correspondence principles, the quantum operators are: x and ħ/i( / x). The commutator can be calculated to be: The consequence is a breakdown of the classical mechanics laws: if there is a complete certainty about the position of the particle (x=0), then there is a complete uncertainty about the momentum (px= ∞). 3. The time and the energy: If a system stays in a state during a time t, the energy of this system cannot be determined more accurately than with an error E. This incertitude is of major importance for all spectroscopies: see Chap 7

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