This proved once and for all the importance of quantum mechanics to the applied sciences and engineering, only 22 years after the publication of the Schrödinger equation. The growing prominence of quantum mechanics in the applied sciences and engineering has already renewed increased research efforts on its mathematical aspects.

## To the Students

The implications and applications of quantum mechanics are limitless, and we are witnessing a time when many technologies have reached their "quantum limit", which is a misnomer for the fact that all methods of classical physics are simply useless in describing or predicting of the behavior of devices at the atomic scale. Quantum mechanics is universal and therefore incredibly versatile, and if you have a sense of mathematical beauty, the structure of quantum mechanics is indeed breathtaking.

## To the Instructor

For these students, it is beneficial to see Bloch's theorem, Wannier states, and basic principles of the theory of covalent bonding embedded with their quantum mechanics course. A proper discussion of photon-matter interactions is therefore also important for a modern quantum mechanics course.

## The Need for Quantum Mechanics

### Electromagnetic spectra and evidence for discrete energy levelsenergy levels

However, once equation (1.1) is accepted as an intrinsic property of electromagnetic waves, it is a small step to make the connection with the line spectra of atoms and molecules and conclude that these line spectra imply the existence of discrete energy levels in atoms and molecules. Schrödinger's merit was that he found an explanation for the discreteness of energy levels in atoms and molecules with his wave equation1(„ h=2).

### Blackbody radiation and Planck’s law

The spectral exitancee.f;T/can also be referred to as the emitted power per area and per frequency unit or as the spectral exitance in the frequency scale. If we measure the energy densitynuŒf;fCf.T/in radiation with a frequency between f andf Cf, then the energy per volume and per frequency unit (ie the spectral energy density in the frequency scale).

### Blackbody spectra and photon fluxes

For a heat source with a temperature of TD5780K, such as the surface of our sun, this gives maxD501nm; c. The number of photons per area, per second, and per unit wavelength emitted by a heat source of temperature.

The photoelectric effect

Wave-particle duality

### Why Schrödinger’s equation?

As a starting point, we recall that the motion of a non-relativistic particle under the influence of a conservative force F.x/D rV.x/ is classically described by Newton's equation. Under the assumption of wave-particle duality, we must assume that this wave function must somehow be related to the wave properties of free particles as observed in the electron diffraction experiments.

### Interpretation of Schrödinger’s wave function

These degrees of freedom are quantum excitations of the vacuum and are mathematically described by quantum fields. 5 Examples of the Schrödinger equation with time-dependent potentials will be discussed in Chapter 13 and following chapters.

### Problems

Why is there usually not much interest in the infinitely many higher order conservation laws (1.26) forn> 1. Equation (1.21) implies that the equation p.t/D mdx.t/=dt from non-relativistic classical mechanics is realized as an equation between expectation values in non-relativistic quantum mechanics.

## Self-adjoint Operators and Eigenfunction Expansions

### The ı function and Fourier transforms

The following is an argument for equation (2.1) and its generalizations to other representations of the function ı. Equation (2.2) is a particularly important realization of equation (2.9) with the normalized sinc function d.x/ D sinc.x/= D sin.x/=x.

### Self-adjoint operators and completeness of eigenstates

Substituting equation (2.20) into (2.19) and (in)formally interchanging integration and summation, we can express the completeness property of a set of functions. However, finally we need only use expansions of the form when evaluating integrals of the formR.

### Problems

However, for many applications of quantum mechanics, it is convenient to use limiting forms of the wavefunctions that can no longer be normalized in terms of equation (2.12), e.g. Use this to find other derivatives of the Fourier representation of a function similar to equation (2.10).

## Simple Model Systems

### Barriers in quantum mechanics

The wave function .x;E1/ is multiplied by the time-dependent exponent exp.iE1t=„/ when passing from .x;E1/ to the time-dependent wave function .x;t/ for movement in the x direction,. A monochromatic wave function can still tell us a lot about the behavior of particles in the presence of a potential barrier V.x/.

### Box approximations for quantum wells, quantum wires and quantum dotsand quantum dots

The energy axis is measured so that in the permitted range the potential energy of the particle disappears, V.x/ D 0, i.e. The energy of the particle is therefore determined by the discrete quantum number n1 and the continuous wave numbers sk2 and k3.

### The attractive ı function potential

48 3 Simple model systems Positive energy solutions of the stationary Schrödinger equation for the ı function potential must have the form. In many situations it is also convenient to use even and odd solutions of the Schrödinger equation.

### Evolution of free Schrödinger wave packets

56 3 Simple Model Systems We can deduce from the example of the free Gaussian wave packet that the kinetic term in the Schrödinger equation drives wave packets apart. Comparison of equation (3.41) with equations gives a constant width of the wave packet ink space and therefore.

### Problems

The wave function .x;t/ can be evaluated numerically from the first line in the following representations. Show that the dispersion of the electron wave packet on the way from the electron gun to the target is completely negligible.

## Notions from Linear Algebra and Bra-Ket Notation

### Notions from linear algebra

For the explicit construction of the dual basis, we observe that the scalar product of the N vector defines a symmetric NN matrix. As any three-dimensional vector, the wave vector displacement can then be expanded in terms of the dual basis vectors according to .

### Bra-ket notation in quantum mechanics

To get some practice with lock notation, let's derive the x-representation of the momentum operator. In superficial terminology this is the statement "the x-representation of the momentum operator is i„r", but the correct statement is equation (4.27).

### The adjoint Schrödinger equation and the virial theorem

0 dt : : : on both sides of this equation then yields the classical virial theorem for the time average hKiT of the kinetic energyKDp2=2m,. 2hKiTD hxrV.x/iT: (4.36) The equation (4.35) applied to ADx implies that the same relationship holds for all matrix elements of the operators KDp2=2mandxrV.x/.

### Problems

Sometimes we seem to violate the symmetry conventions in the Fourier transforms of the Green's functions, which we will encounter later. Show that the free propagator is the representation of the one-dimensional leisure evolution operator.

### Uncertainty relations

Minimal values of the uncertainty A with which the observable Ao can be measured are directly related to the commutator of the operator A with other operators. Equation (5.1) implies for the operators x and p for location and momentum of a particle Heisenberg's uncertainty relation.

### Frequency representation of states

The sum over continuous indices˛ will include an integration overı.!E˛=„/ for E˛ in the continuous parts of the spectrum of H, and although the Fourier transform from j .t/ito a frequency-dependent condition does not exist in the sense of classical Fourier theory,j .!/exists as a sum ofıfunctions over the discrete spectrum of the Hamiltonian plus a sum over continuous states. A major source of error is to confuse the frequency representationj .!/i of a state, which exists only in the distribution sense, with energy eigenstatesj ˛iofHhj ˛i DE˛j ˛i in the sense of classical analysis.

### Dimensions of states

The wave functions hxji and hkji therefore have dimension of length1=2 or dimension of length1=2, respectively, while the representations or wave functions shxjkiorhxjk;˙i are dimensionless. In three dimensions the statesjn1;n2;n3iin a cubic quantum dot are dimensionless while the statesjn1;n2;ki(3.10) in a cubic quantum wire have the dimension length1=2 in accordance with their completeness relation.

### Gradients and Laplace operators in general coordinate systemscoordinate systems

These equations are particularly convenient when the new coordinates are given in terms of Cartesian coordinates xi,˛D˛.x/. We must take this into account when calculating the Laplace operator in the new coordinate system.

### Separation of differential equations

The eigenstates of the separable Hermitian differential operators will be automatically factorized into the eigenstates of the corresponding low-dimensional operators. The non-degeneracy of the one-dimensional eigenvalues EQ1 and the linear independence of the corresponding eigenstates therefore imply.

### Problems

Equation (5.26) is more convenient than (5.23) for calculating the Laplace operator in parabolic coordinates. In the flat spaces that we deal with in this book, equation (5.40) is the condition for a straight line in terms of the curvilinear coordinates˛.

## Harmonic Oscillators and Coherent States

*Basic aspects of harmonic oscillators**Solution of the harmonic oscillator by the operator methodby the operator method**Construction of the states in the x-representation**Lemmata for exponentials of operators**Coherent states**Problems*

We use (6.15) in the second step of the calculation below (and then 1 additional step to arrive at the final result). Oscillator energy eigenstate space representations can be constructed in the same way as x representations.

## Central Forces in Quantum Mechanics

### Separation of center of mass motion and relative motion

The separation of the center of mass motion in the present form works for any potential V.r/which depends only on the separation vector of the two particles. If we model the potential well through a three-dimensional oscillatory potential, the center of mass motion can be described by the eigenstates of the oscillator.

The concept of symmetry groups

### Operators for kinetic energy and angular momentum

2 r2hrjM2j i DEhrj i; (7.12) and we can deal with the angular part in the equation by first solving the eigenvalue problem for the operator M2. A useful tool for the analysis of angular momentum operators is the symmetry of the Eddington tensor.. and normalization implies that nŠ=2 of the components have the value 1, nŠ=2 of the components have the value 1, and nnnŠcomponents vanish.

### Matrix representations of the rotation group

It also implies that the direction 'O of the vector' is the direction of the axis of rotation. It is a consequence of the general Baker-Campbell-Hausdorff formula in Appendix E that the combination of any two rotations into a new rotation is completely determined by the commutation relations (7.21) of the generators of rotations.

### Construction of the spherical harmonic functions

Dı.cos#cos#0/ı In bra-ket notation we can write these completeness relations for functions on the sphere as . Knowledge of a continuous functionh#; 'jfion de sphere is equivalent to knowing the innumerable numbers sh`;mjfi.

Basic features of motion in central potentials

2.kr/Y`;m Our conventions for the phase and the normalization of the radial wave function are motivated by the expansion of plane waves in the form of spherical harmonics.

### Bound energy eigenstates of the hydrogen atom

For the meaning of the radial wave function, remember that the complete three-dimensional wave function is. This means that n2;`.r/ is a radial profile of the probability density j n;`;m.r/j2 to find the particle (or rather the quasiparticle describing the relative motion in the hydrogen atom) in the locationr, but note on that in any specific direction.#; '/the.

### Spherical Coulomb waves

In addition to normalization, the spherical Coulomb waves k;`.r/become radial bounded state wavefunctions n;`.r/through the substitution ik!.na/1. To construct such a state from spherical Coulomb waves, we can use that equation (7.51) which tells us the decomposition of the plane wave exp.ikz/in terms of free states of sharp angular momentum (7.48).

### Problems

We cannot construct the energy eigenstates of the hydrogen atom, which separate in the coordinates xe and xp of the electron and proton. What velocities would equation (7.81) predict for the velocities of the proton and electron in the ground state of the atom.

## Spin and Addition of Angular Momentum Type Operators

*Spin and magnetic dipole interactions**Transformation of scalar, spinor, and vector wave functions under rotationsfunctions under rotations**Addition of angular momentum like quantities**Problems*

We will superficially denote all these operators (including spin) simply as angular momentum operators in the following. Therefore, we can construct a third state in the combined angular momentum basis, which is orthogonal to the other two states.

## Stationary Perturbations in Quantum Mechanics

### Time-independent perturbation theory without degeneracieswithout degeneracies

172 9 Stationary Perturbations in Quantum Mechanics We know the unperturbed energy levels and eigenstates of the solvable HamiltonianH0. Ei.0/E.0/k : (9.8) States in the continuum part of the H0 spectrum will also contribute to shifts in their energy levels and states.

### Time-independent perturbation theory with degenerate energy levelsenergy levels

Summary of the first-order displacements of the level E.0/i if the perturbation raises the degeneracy of the level. The projections of the first-order shifts of the eigenstates onto states in other sectors of the degeneracy are.

### Problems

Find an approximation H0 for the Hamiltonian of the atom, where you can write down exact energy levels and eigenstates of the atom. Use the remaining expressions inHH0 to calculate the first-order corrections to the atom's energy levels and eigenstates.

## Quantum Aspects of Materials I

Bloch’s theorem