Elementary Quantum Mechanics

This is the first graduate course on elementary quantum mechanics in Internet written in Romanian for the benefit of Romanian speaking students (Romania and Moldova). It is a translation (with corrections) of the Spanish version of the course (physics/9808031, English translation physics/0004072), which I did at the request of students of physics in Bucharest. The topics included refer to the postulates of quantum mechanics, one-dimensional barriers and wells, angular momentum and spin, WKB method, harmonic oscillator, hydrogen atom, quantum scattering, and partial waves


INTRODUCTION
The foundations of non-relativistic wave mechanics, the branch of quantum theory to be treated here, were laid down by Schrödinger in 1926.Shortly before this, Heisenberg had formulated a quantum mechanics based on matrix algebra.His theory was soon shown to be equivalent to Schrödinger's, but for many practical purposes wave mechanics proved easier to work with than matrix mechanics.
In a few years this new quantum mechanics was developed to a high degree of perfection by a large number of physicists, among whom Bohr, Born, and Heisenberg were particularly active in formulating a consistent physical interpretation of the mathematical apparatus of the theory.In 1927 wave mechanics was generalized by Pauli to apply to particles with spin, and in 1928 Dirac proposed the relativistic wave equation for the electron.
The new quantum mechanics dating from 1925-26 replaced the so-called old quantum mechanics which had its roots in Planck's introduction, at the turn of the century, of the concept of quanta in physics.In order to explain the frequency distribution in heat radiation from solid bodies, Planck was obliged to assume that radiation was transmitted in quantum units, each containing the energy v being the frequency of the radiation emitted, while h is Planck's universal constant, having the dimension of an action (energy × time), and magnitude h = 6.6 x 10-27 erg sec. (1.1.2) This quantum description of electromagnetic radiation, supported by Einstein's interpretation of the photoelectric effect (1905), was utilized by Bohr (1913) in a theory capable of describing the line spectrum of hydrogen on the basis of Bohr's famous postulates.This formed the basis of the 'old' quantum mechanics which was then developed by Bohr, Sommerfeld, and others to explain the discrete energy states of the atoms and to give a qualitative description of atomic spectra and of the periodic system of chemical elements.
The old quantum mechanics* was essentially a classical mechanics, subject to restraints, the so-called quantum conditions, which select certain of the infinitely many possible classical orbits as the only permissible ones.The rules of quantization could be applied only to systems classically performing periodic movements such as, for example, classical elliptical orbits of an electron in a Coulomb field, or harmonic vibrations of atoms about their mean position within a molecule.On the other hand, the theory could not be made to apply to systems which change in time, as in radioactive decay or collisions between atomic particles.Hence the old quantum mechanics could by no means be considered a final and consistent theory.
Before proceeding to the treatment of wave mechanics and its results, it is worth mentioning how it was possible to link wave concepts with material particles.In 1923 the Compton effect was discovered; it showed that in the case of scattering of short-wave electromagnetic radiation by electrons, an electromagnetic wave should be considered as a stream of particles, each having the energy E = hv and a momentum in the direction of propagation of the wave of magnitude h being Planck's constant and λ the wavelength.In this case the radiation quanta are particles of mass zero (photons).**Hence in certain experiments electromagnetic radiation should be regarded as a stream of particles, whereas, on the other hand, the pronounced interference effects in the case of the passage of X-rays through crystals (Bragg and Laue diffraction) show that here the electromagnetic radiation occurs in the form of waves.
In 1924 de Broglie proposed, as a hypothesis, that the above wave-particle dualism in the description of electromagnetic radiation should be extended to apply also to material particles, in such a way that it should be possible to associate with a particle a wave of frequency and wavelength determined by, respectively, the energy and momentum of the particle as in (1.1.1)and (1.1.3).This hypothesis was supported experimentally in 1927, when Davisson and Germer, by scattering electrons of known momentum on a nickel crystal, found that the electrons emerged in precisely the directions where one would expect to find intensity maxima for the diffraction of X-ray radiation of a wavelength depending on the momentum of the electrons as in (1.1.3).
This suggests that there is an analogy between wave mechanics for material particles and Maxwell's wave theory for electromagnetic radiation.Although the analogy can be pushed too far (for one thing, photons have zero mass and cannot therefore be described in non-relativistic terms), it may serve to hint at the way in which wave mechanics differs from classical mechanics.The analogy will be used in the following, where an attempt is made to present some plausibility arguments for the basis of wave mechanics.

Problem
1.1.1Calculate the de Broglie wavelength for electrons with an energy of 100 eV, as well as for particles with mass 1 g and velocity 1 m/sec.

THE SCHRÖDINGER EQUATION FOR A PARTICLE
Quantum mechanics is a mathematical formalism, which, on the basis of knowledge of the state of a system, at a given time, allows the calculation of the state of the system at a later time, so that the result of experiments can be predicted.In quantum mechanics, the state of a particle is described by a wave function.At any given moment, the wave function describes a certain field in space, associated with the immediate state of the particle.The wave function develops in time in a manner depending on the environment of the particle, and the state of the particle at a later time is then usually described by another field in space.
In order to use this type of formalism it is necessary to translate experimental knowledge of the physical state of particles into mathematical properties of the wave function and vice versa, and one must also be able to calculate the development in time of a given wave function.For the latter purpose we use a differential equation, the Schrödinger equation, which governs the behaviour of the wave field in time and space and hence occupies a position analogous to that of Maxwell's equations within the electromagnetic field theory.
As to the association of a wave function with a given physical situation, there is a decisive difference from the electromagnetic analogy.In fact, while the electromagnetic fields, the electric and magnetic field strengths, are directly measurable physical quantities, the quantum mechanical wave function is an auxiliary concept with less direct connection with the experiments.
It is possible to present quantum mechanics in an axiomatic form by postulating the Schrödinger equation and a set of rules for the physical interpretation of the wave function.However, in the following we shall use a less systematic approach, introducing the different aspects of the formalism by degrees, and trying to render the assumptions plausible.
We shall take Davisson and Germer's experiment as our starting point, and we shall further place the rather natural demand on our wave description that in the limit-at least for macroscopic phenomena-it must agree with Newtonian mechanics.For a detailed description of the Davisson and Germer experiment, and of other diffraction experiments with material particles (electrons as well as atoms and molecules), see, for example, H. Semat, Introduction to Atomic Physics.
The essence of these experiments may be summed up in the following way.When a beam of atomic particles of well-defined momentum p hits a crystal, scattered particles are observed mainly in certain directions; these are distributed in such a way that when the particles are registered on a photographic plate, it shows an interference pattern perfectly analogous to the one which would result if, in the experimental arrangement, the incident beam of particles were replaced by monochromatic X-rays of wavelength λ = h/|p|.This relation between the momentum of the particles and the wavelength determined by the interference picture is particularly accurate only for fast particles.We shall return to the significance of the deviations; at the moment we ignore such elaborations.
Having made this reservation, one may 'explain' the electron diffraction experiments by formally associating the beam of particles with a plane harmonic wave train proceeding in the direction of the beam of particles, and having a wavelength which depends on the momentum of the particles in the above manner.
There is, of course, no necessity for the associated wave to be of an electromagnetic nature; but it must be possible to imagine that it is diffracted in crystals.The only necessary condition for this is that the incident wave train be in some way disturbed by the strong force fields near the atoms of the crystal lattice so as to give rise to scattered waves oscillating at the same frequency as the incident wave.Furthermore, since interference must be possible, we must assume the validity of the superposition principle for the associated waves, in the same manner as for electromagnetic fields.These two assumptions are the only ones needed to arrive at the Bragg condition for the directions in which the scattered waves from a large number of lattice points are added in phase, and therefore combined to a scattered wave of large amplitude.
The amplitude of the scattered waves hitting the various points of the photographic plate must in one way or another be interpreted as a measure of the flow of electrons in this region of space, in the same way as the intensity of the electromagnetic waves determines the flow of photons in the X-ray diffraction experiment.However, in our preliminary considerations we need not worry about the quantitative relations between the associated wave field and the distribution of the electrons in space.It is sufficient to assume that the electrons occur mainly where there are waves of large amplitudes, whereas no electrons are found where the waves cancel each other.
Before continuing the discussion of the associated waves we shall consider wave fields in general.A wave function describing a plane harmonic wave may be expressed in a number of ways, e.g. by either of the following three representations where A is a constant (the amplitude).
The above field quantity, ψ, might be a component of the electric field strength.In this case the complex formulation (1.2.1c) is only used for convenience, the field strength being represented by the real part.
In the expressions (1.2.1) it is seen that the phase, k ⋅ r -ω t, at a given time is constant on any plane normal to k, since such planes are represented by k ⋅ r = constant.The vector k, which is called the wave vector, is hence normal to the wave plane.The phase difference between two wave planes at a mutual distance A is |k| λ.If λ is to represent the wavelength, the phase must increase by 2π over this distance, and hence the magnitude of the wave vector (the wave number) * must be given by (1.2.2) If the x axis is chosen in the direction of k, the phase may also be written in the form kx-ω t, or 2π (xlλvt), where v = ω /2 π.
Over a period of time, ψ oscillates at a fixed point in space with the angular frequency ω.At a fixed point with abscissa x, the phase decreases over a short period of time δ t by the amount ω δ t, but this is compensated for if we simultaneously increase x by the quantity δ x = (ω / k) δ t.Thus the wave planes with a given phase proceed in the direction of k with the phase velocity (see Fig. 1.1) It is possible to observe neither the angular frequency ω nor the phase velocity in diffraction experiments which determine only the wave vector k.For electromagnetic waves the frequency can be found from the wavelength since the waves comply with the Maxwell equations, leading to a wave equation of the type where ∆ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 is the Laplace operator, while c is the velocity of light.
arbitrary linear combinations of functions which are themselves solutions, and in this way it is possible to form new solutions of (1.2.4) by the superposition of harmonic waves having the proper relation between ω and k.
In order to derive the wave equation for our beam of particles, we shall proceed in the opposite manner.From the demand for correspondence with the Newtonian mechanics we shall first find the relation between k and to which must be obeyed by the superposition of the harmonic waves, and then see which differential equation is satisfied by these waves.
By the superposition of harmonic waves with different wave-lengths it is possible to form a wave function which has large amplitudes only in a small region of space, where the waves are chosen to have the same phase, whereas they tend to cancel elsewhere.At first we try to superpose waves in one dimension with wave numbers in a small interval between kδ k and k + δ k.Choosing a complex formulation we get for the resulting wave function at the time t = 0  It is seen that the wave function essentially vanishes outside a region whose width is inversely proportional to the width 2δ k of the interval from which we have chosen our wave numbers.Here we have superposed harmonic waves with equal weight from the entire interval.By choosing different weight functions it is possible to change the shape of the wave packet, but its extent cannot be reduced below the order of magnitude of where δ k represents the width of the wave number interval which contributes materially to the wave packet.
The relation (1.2.5) may be understood qualitatively by considering that harmonic waves which are in phase in the centre of the wave group, must have a difference in wave number of δ k in order to get into counterphase at the distance δ x from the centre, and that consequently δ k δ x ≈ π.
Similarly, one may construct a wave packet in space by superposition of harmonic waves with wave vectors k', which deviate slightly both in magnitude and direction from a fixed vector k.
where we integrate over all k' space with a weight factor g(k'), which is vanishingly small outside the domain |k x -k' x | .δk, |k y -k' y |.δk and |k z -k' z |.δk.
The linear extent of the wave group in space can then be of the order of magnitude of 1/δ k in all directions.
The development in time of this type of wave packet is determined by the frequency of the harmonic waves incorporated in the wave packet, and it is necessary here to remember that ω is a function of the wave number.The wave packet (1.2.6) is at the time t then given by (1.2.7) We now exploit the fact that the wave packet includes only wave vectors k' very close to k, and that therefore ω in the integrand will deviate only slightly from the value ω (k).To first approximation we then find ω (k') = ω (k)+grad k ω ⋅ (k'-k), discarding terms of second and higher order in the quantities k' x -k x , k' y -k y , and k' z -k z .
Equation (1.2.7) may then be written in the form In this approximation the integral in (1.2.8) is evidently a function of r-t ∇ k ω , and consequently the wave packet proceeds with the velocity v g = ∇ k ω. (1.2.9) This so-called group velocity for the waves is determined by the dependence of frequency on wave number, and it will therefore usually be different for wave packets of different 'mean wave vectors' k.Fig. 1.3 gives an idea of the movement of a one-dimensional wave packet.Since phase velocity usually differs from group velocity, we have to imagine that wave peaks are being displaced within the group, i.e. that new peaks develop continually, while others disappear.It should be noted that the approximation leading to (1.2.9) is usually good only for a limited time, since the discarded terms in the phase are no longer insignificant compared to π when sufficient time has elapsed.Hence over long periods of time the wave packet may change its shape and become flattened [see problem 1.2.1].
For electromagnetic waves in vacuum we have ω = c k, which implies that phase velocity and group velocity are equal.In this case the above approximation is exact for a one-dimensional wave group, which hence will be transmitted in unchanged form.Similar relations apply to sound waves, as may be concluded from the fact that music is heard without distortion at any distance from its source.However, in the case of surface waves of liquids there is a more complicated relationship between ω and k.
We now return to considering the waves which are to be associated with the incident beam of particles in the Davisson and Germer experiment.If we assume that the electrons have momentum p, we have seen that we must assign to them a plane harmonic wave train with a wave vector in the direction of p of magnitude k = 2π /λ = 2π p / h.This may be written k = ¬ -1 p, (1.2.10) where we have introduced the commonly used abbreviation ¬ = h/2 π.By superposition of such harmonic waves with wave vectors deviating only slightly from k, and therefore, according to (1.2.10), all associated with particles of momenta virtually identical with p, we can build up a wave function representing a wave packet.This wave function differs from zero only within a small domain of space, and it must therefore be possible to link it to a particle present in this domain and with a momentum ≈ p.If we desire to reduce the domain to linear dimensions of an order of magnitude a, it is necessary to superpose wave vectors with components in intervals of the order of magnitude 1/a, which, according to (1.2.10), corresponds to a distribution of the components of the momentum within an interval of a minimum order of magnitude of ¬/a.However, in the case of macroscopic particles, this 'haziness' in the momentum will be totally imperceptible.
If, for example, we imagine a particle with a mass 1 g, and localized with the high accuracy of 10 -4 cm, the construction of a wave packet of these dimensions will cause a lack of definition or spread in the velocity components of approximately ¬/am ≈ 10 -23 cm/sec.In order to make our wave description compatible with classical mechanics, it is apparently necessary to demand that the above wave packet has the velocity v = p/m.We must therefore insist that this is the group velocity for waves with wave vector k = p/¬.By using the expression (1.2.9) for the group velocity we then find ∇ k ω = p/m = ¬k/m, which may immediately be integrated to give ω = ¬k 2 /(2m) + C, (1.2.11) where C is an arbitrary constant.Hence an arbitrariness is permissible in the absolute magnitude of the frequency (but not in its dependence on k), causing, according to (1.2.3), an arbitrariness in the magnitude of the phase velocity.This presents no difficulty, since in the case of associated waves the phase velocity and the frequency need not be observable quantities.This is connected with the fact that the wave function, contrary to the case of electromagnetic fields, is not accessible to direct measurements.
We can now construct a wave equation for our associated waves.In order to allow superposition of the solutions, the equation must be linear.Using (1.2.11), and for convenience putting C = 0, we can express the phase of the harmonic waves which are to be the solutions of the desired differential equation thus: (1.2.12) We may then choose between the various possibilities for harmonic wave functions (1.2.1) which, with the phase (1.2.12), are Guided by the form of the electromagnetic wave equation (1.2.4), we find by direct calculation that ∆ψ = −k 2 ψ for all four wave functions.However, if we take the partial derivative of ψ with respect to time, we see immediately that forms III and IV distinguish themselves by the fact that already the first derivative is of the form: (constant) × k 2 ψ, which offers the possibility of a particularly simple wave equation.If wave function III is chosen, we find that ∂ ψ /∂ t = -i(¬/2m) k 2 ψ, and we may then use the wave equation This differential equation is the Schrödinger equation for free particles of mass m.Its harmonic solutions are of the form The choice of the wave function III in preference to IV is merely a convention.The choice of IV would only cause a change of sign on one side of (1.2.13).On the other hand, wave functions of types I and II would require a differential equation of the fourth order in the space coordinates.The simple form (1.2.13) admittedly requires the use of complex wave functions, but this only means that we work with two real functions at once, the real and the imaginary part of ψ.Thus, when we say that the state of a particle is described by a wave function, then we actually mean a set of two real functions * .It may also be noted that the differential equation (1.2.13) is of only first order in time, and that consequently the dependence of the wave function on the space coordinates at a certain time determines ψ (r, t) for all times.
Before we move on to a discussion of the interpretation of the wave function on the basis of the Schrödinger equation, we shall establish the Schrödinger equation for a particle in a conservative force field.
Here we might extend our considerations concerning free particles, attempting to construct wave packets moving in the classical manner in macroscopic force fields, in order to arrive at the wave equation.However, in order to ease our progress, we will make a few more guesses.
We observed previously that in the case of the Davisson and Germer experiment the relation (1.2.10) between the momentum p of the incident electrons and the associated wave vector k holds strictly only for sufficiently fast electrons.In order to explain the interference picture with decreasing kinetic energies, T = p 2 /2m, for the electrons, it became necessary to introduce into the Bragg formula a slightly smaller wavelength than h/p, corresponding to a wave number slightly higher than p/¬.For nickel crystals the variation with T of the corrected wave number k' could be repre-sented by a formula This result might still be interpreted in accordance with the de Broglie hypothesis, if only it were assumed that the electrons move with a slightly higher momentum T eV T p p 21 ' + = within the crystal than outside.This type of effect might actually be expected, since energy is required to remove an electron from the crystal.Inside the crystal the electron is, on the average, affected equally in all directions by the surrounding atoms, but at the surface the actions add to an approximate jump in the potential for the electron.
By using T = p 2 /2m, one finds that the above equation for p' is equivalent to which may be regarded simply as the energy integral for the electrons, assuming that the potential energy is Φ 0 = -21 eV inside the crystal and Φ = 0 outside.In actual fact these considerations do not teach us anything new about our associated waves.The relation (1.2.15) merely determines how, for a given external energy T of the electrons, we may calculate the magnitude of the wave vector in (1.2.14), in order that this wave function may be associated with the electrons inside the crystal.However, the Davisson and Germer experiment also teaches us something about the wave function outside the crystal.It might be hoped that there we have a de Broglie wavelength corresponding to the electron momentum, p, outside the crystal.If this is the case, we have waves of different wavelengths outside and inside, and one may then expect refraction of a plane wave train which does not strike the surface at a right angle.This must, of course, be taken into consideration when comparing the external interference picture and the diffraction angles calculated on the basis of the Bragg conditions within * In relativistic quantum mechanics, wave functions with even more components are used.the crystal.It was indeed found in the Davisson and Germer experiment that in order to obtain full agreement with the observed interference picture one must assume a refractive index sin ϕ / sinϕ' = λ /λ ' (1.2.16) between crystal and vacuum.
Here ϕ and ϕ ' are the angles between the normal of the crystal surface and the wave normals outside and inside the crystal respectively, while λ and λ ' are the de Broglie wavelengths corresponding to the electron momenta in the two regions.An elementary consideration (see Fig. 1.4) shows that the refractive index is the Fig. 1.4 ratio between the phase velocities of the two regions, so that sin ϕ / sinϕ' = v p /v p '.
From (1.2.3) and λ /2π = 1/ k, one finds the refractive index (1.2.16) when one assumes identical frequencies in the two regions.
We have now learnt how to construct our associated wave function when we want to describe electrons moving with a given total energy in two domains of different, but constant potentials.
In the two domains we need different wave vectors, corresponding to the electrons' classical momenta at constant total energy, but we need the same frequency.The latter applies also to all other wave phenomena and is a necessary condition if we are to combine the wave functions in the two domains in such a way that they remain continuous in time.Otherwise the combined wave function could not be a solution to a differential equation in space and time.Apparently the demand for identical frequencies with different wave vectors is not satisfied by the expression (1.2.12) for the phase, which followed from (1.2.11) with C = 0.However, we may manage by suitable adjustment of the arbitrary constant C in the frequencies for the two domains.Within these the wave vectors are known from k = p / ¬ and k' = p'/ ¬.If we let the potential energies be Φ 0 and Φ' 0 , and the total energy be E, we find from (1.2.11) for the frequencies of the two domains The desired result is then obtained by putting C = Φ 0 / ¬, C' = Φ' 0 / ¬, whence the frequency in both domains is given by the simple expression ω = E / ¬. (1.2.17) The previously mentioned arbitrariness in the absolute magnitude of the frequency now corresponds perfectly to the freedom of choice of the zero point for potential energy.
The inclusion of the extra term, C = Φ 0 / ¬, in (1.2.11) causes the presence of an extra term, -Φ 0 / ¬ t, in the phase (1.2.12).The wave function exp (iϕ) is no longer a solution of (1.2.13), but satisfies instead the differential equation where Φ(r) represents the potential energy at the point determined by the vector r.Obviously this differential equation comprises (1.2.18) and (1.2.19), since we assumed Φ(r) = Φ 0 and Φ(r) = Φ ' 0 for the two domains.
Similarly (1.2.20) may be seen to lead to (1.2.13) when, for a free particle, we make the usual assumption Φ(r) = 0 for all r.
We shall henceforth assume that a particle with a mass m, in any conservative force field with potential energy Φ (r) may be associated with a wave function ψ (r, t), which is a solution to (1.2.20).This differential equation is called the Schrödinger equation for particles with mass m in the field Φ (r).However, our arguments for the validity of (1.2.20) are applicable only for the specific case of Φ being piecewise constant in space.In the more general case the Schrödinger equation is, of course, not satisfied by wave functions representing plane harmonic wave trains or wave packets built from them.Whether or not the solutions provide correct descriptions of natural phenomena in such cases can be decided only by experimental tests of the deduced results.
In the following, we base wave mechanics on the Schrödinger equation (1.2.20).Subsequently we shall show that it leads to the Newtonian laws of motion for particles in macroscopic force fields.Problems 1.2.1 Write down the expression corresponding to (1.2.7) for development in time of a one-dimensional wave packet.Use the expression (1.2.11) for the frequency, and show that the shape of the wave packet may be considered unchanged only for a duration which is small compared to ma 2 / ¬, where a represents the order of magnitude of the extent of the wave packet.Find this critical time for the case of m= 1 g, a = 10 -4 cm.where p and E represent corresponding values of the classical momentum and total energy for particles of mass m.

INTERPRETATION OF THE WAVE FUNCTION
Before we can begin deriving quantitative results from the Schrödinger equation we need more well-defined ideas on the translation of the properties of the wave function into specifications as to the physical state of the particles.In the course of the discussion on electron diffraction experiments it was mentioned that the wave function ψ (r, t) should give a certain measure of the occurrence of particles at r at the time t.A simple, non-negative quantity which may be formed from the complex wave function is the absolute square |ψ | 2 = ψ *(r, t) ψ (r, t), and it niax.therefore appear to be reasonable to let |ψ (r, t)| 2 tentatively represent the density of particles at time t at the point r.According to this assumption, then, it is the variation in space of |ψ | 2 which determines the distribution of darkening on the photographic plate in a diffraction experiment.However, the characteristics of the interference picture turn out to be independent of the intensity used.Even if the intensity is so low that only one electron at a time passes the crystal, the same interference picture will be built up over a long period of time as by brief irradiation of high intensity.Hence the interference effects are not dependent on the presence of a large number of electrons in the crystal and cannot be ascribed to an interaction between particles in the incident beam.Consequently the wave function must be associated with the single electron, and a reassessment of our interpretation of |ψ | 2 as a density is necessary.

Fig
Fig. 1.1 ψ(x) represents a wave group, also called a wave packet, as indicated in Fig.1.2.