Quantum Physical Phenomena in Life (and Medical) Sciences

Quantum Physical Phenomena in Life (and Medical) Sciences Quantum Physics Experiments and principles beyond the capacity of the classical (Newtonian) physics Péter Maróti Professor of Biophysics, University of Szeged, Hungary Suggested text: S. Damjanovich, J. Fidy and J. Szőlősi: Medical Biophysics, Semmelweis, Budapest 2006

New (quantum) physics is needed to describe and to understand • the chemical-physical nature of electrons in atoms and molecules (bonds, interactions, etc.), • the behavior of nucleus (radioactivity, nuclear reactions, radiation therapy, etc.), • the energetic changes in nuclea, atoms and molecules (spectroscopy) and • the image formation from human body and organs (iconography), because classical physics fails to work in these fields of the microworld.

What is new that quantum physics offers? 1. Some physical quantities are quantized - the energy of electromagnetic fields (black body radiation) - the energy of the atoms (absorption and emission spectra of atoms, Franck-Hertz experiment) - the magnetic moments of particles, the spin (Einstein-de Haas experiment) 2. Wave-particle dualism - Wave properties of particles (Davisson-Germer experiment, Tunneling) - Particle properties of waves (photoelectric (Hallwachs) effect) 3. „Unusual” distribution - Spatial distribution: description using the term of probability (wave function) - Deviation from equipartition (temperature-dependence of the molar heat capacity of solids) 4. Uncertainty principle after Heisenberg Complementer quantities cannot be measured simultaneously with arbitrarily small precision.

1. The energy is quantized: the electromagnetic field and the atomic oscillators • Black body radiation and the UV catastrophe • explained by quantized energy levels for light and emitting oscillators (Planck). Light emitted by a black body has a spectrum that depends on temperature. The “ideal black body” is a box with black walls, opening to the exterior through a small hole. Radiation escaping through the hole will be in thermal equilibrium with the temperature of the walls.

Classical physics attempted to explain the shape of the curve of power (or energy density, ρ) as a function of wavelength through four laws. 1. Kirchhoff’s law: e/a = E(λ,T), for any substances, where e: emission and a: absorption. For black body: a = 1. 2. Wien’s displacement law: T·λmax= 2896 μm·K (constant) T is the absolute temperature andλmax is the peak of the curve. 3. Stefan-Boltzmann law: Etotal = σ·T4 Etotalis the emittance (total power per unit area), and σ= 56.7 nW·m-2·K-4. This is why bulb filaments are run hot! 4. Rayleigh-Jeans law: The first three laws worked fine, but not the Rayleigh-Jeans law that led to „UV catastrophe”

In the UV, the Rayleigh-Jeans curve diverges to infinity Ultraviolet catastrophe Rayleigh-Jeans curve Experiment black body radiation All energy levels allowed, so fraction of energy density with high energy increases as temperature increases. The radiation escaping from a black body is determined by the energy loss from vibration of molecules (the electromagnetic oscillators) in the walls. These increase with T. The energy of the light emitted is determined by the oscillators.

Planck proposed that the energy of the electromagnetic oscillators was limited to discrete values, rather than continuous. Planck’s famous equation is E = nhn, where n = 0, 1, 2, …, n (= c/l) is the frequency, and h is Planck’s constant, 6.626 10-34 Js. Planck’s curve The energy levels are quantized. At any temperature, the transitions at higher energy levels are less likely to occur, so contribution to energy density falls off in high energy range (UV). As the energy (E = hn = hc/l)approaches kT, the term in brackets approaches 1, and the Planck equation becomes the same as the Rayleigh-Jeans equation. Conclusion: the energy of the oscillators that make up the electromagnetic field are QUANTIZED

2. Temperature dependence of molar heat capacities of solids The heat capacity (specific heat) is the proportionality factor relating temperature rise, DT, to the heat applied: q = C·DT. The classical view was that C was related to the oscillation of atoms about their mean position, which increased as heat was applied. If the atoms could be excited to any energy, then a value of C = 3R = 25 J·K-1 was expected, and this value, proposed by Dulong and Petit, was observed for many systems at ambient temperature. However, the expected behavior was a constant value as a function of T, and this was not seen, and some elements (diamond) had values way off. Upon lowering the temperature, the observed heat capacity did not remain constant but approached to zero.

Temperature dependence of molar heat capacities of solids The Dulong and Petit’s law comes from the equipartition theorem of the classical physics. Significant deviations were observed at low temperatures. Einstein and Debye formula: The atoms oscillate with the same frequency (Einstein) or with the range of frequencies (Debye).

Temperature dependence of molar heat capacities of solids Einstein showed that if Planck’s hypothesis of quantized energy levels was applied, the equation: provided a good fit. This was later improved by Debye who allowed a range of values for n. Conclusion: the energy of atomic oscillators in matter are also QUANTIZED.

Conclusions from Planck’s solution to the UV catastrophe and Einstein’s solution to the heat capacity problem • Heating a body leads to oscillations in the structure at the atomic level that generate light as electromagnetic waves over a broad region of the spectrum. This is an idea from classical physics. • From Planck’s hypothesis, the properties can be understood if the energy of the oscillations (and hence of the light) are constrained to discrete values, - 0, hn, 2hn, 3hn, etc. At any frequency value, the intensity of the light is a function of the number of quanta, n, at a fixed energy, determined by hn. This is in contrast to the classical view in which energy levels were assumed to be continuous, and intensity at any frequency was dependent on the amplitude of the wave. • A similar conclusion comes from Einstein's treatment of the heat capacity, but here the effect is seen from the oscillations generated in the substance on application of heat. Absorption of energy is therefore also quantized.

1. The energy is quantized: the atoms Lines of hydrogen emission spectrum When a high voltage is discharged through a gas, the atoms or molecules absorb energy and from collision with the electrons, and re-emit the energy as light. It is found that the light is not a continuous range at all frequencies, but is constrained to a few narrow lines (right). Explanations of the emission spectrum of atomic hydrogen played a critical role in development of a quantum mechanical understanding of the structure of atoms. Lines similar to those in the hydrogen emission spectrum were seen as absorption lines in the light from stars.

Balmer studied the emission spectrum in the near UV-visible region, and noticed that the distribution of the lines along the wavelength scale showed an interesting pattern that could be described by the Balmer formula: where n is 3, 4, 5,… Later work revealed a more extensive set of lines in the UV and IR, which all followed the general pattern given by: where is the Rydberg constant, and nL and nH are lower and higher value integers.

Term scheme of the H-atom Balmer series:

Franck-Hertz experiment One of the most demonstrative experiments for the quantization of energy of atoms. The wavelegth of the emitted light is The steps of the current-voltage characteristics describe the set of discrete energy levels (term scheme) of the mercury atom. At the first step of U = 4.9 V, λ= 253.7 nm

1. The magnetic moment is quantized: Spin; The Einstein- de Haas Experiment An iron bar (S) is hung from a string (R) in such a way that it can rotate around its axis. From the torsion vibrations of the string, the angular momentum N, which the bar gets as a whole when it is magnetized by external currents through the spoils (E), can be measured. Suppose that n electrons with their elementary angular momentum j contribute to the magnetization. m: mass of the electron, q =-e: elementary charge of the electron, c: speed of the light, μB: Bohr magneton As By measuring the macroscopic quantities N and Mbar, we can determine the gyromagnetic ratio gof the elementarymagnetic moments(the Landé factor): g = 2. The magnitude of two Bohr magnetons for the magnetic moment excludes the orbital magnetic moment as the source of the ferromagnetism and can be only explained by the existence of the electron spin.

Deflection of Ag atom beam by a magnetic field: theStern-Gerlach experiment Splitting of Na D-line emission by a magnetic field: the Zeeman effect. Classical physics: continuum of deflections No magnetic field, no splitting N S Quantum physics: two sharp peaks separated • The atomic beam passes through an inhomogeneous magnetic field. One observes the splitting of the beam into two components. Consequences: • Experimental demonstration of directional quantisation and • The direct measurement of the magnetic moments of atoms. Some electron energy levels could be split by a magnetic field. This is the Zeeman effect.

The electron spin quantum number, and Pauli’s exclusion principle. The interpretation of the spin by spinning the electron around the axis is a fiction only that makes the imagination easier (but it is not true)! In order to account for the magnetic splitting of atomic lines, it was necessary to understand how the magnetic properties of the electron could be included in the grand scheme of the quantum physics. spin up or a spin down or b Since the electron has a negative charge, it is clear that any electron moving in an orbital should also generate a magnetic field. Pauli realized that the problem could be inverted, - why do all atoms not show a magnetic response? The answer he proposed was that electrons normally came in pairs, so that their magnetic effects cancelled out. This would be the case if, in addition to movement in an “orbit”, the electrons were also spinning on their axes. Each electron would then be a magnet. A pair of electrons in the same orbit would cancel out, but only if they had opposite spins. More formally, Pauli’s exclusion principle states that two, but no more than two, electrons can occupy any orbit. As a result, the number of electron orbitals was doubled, by adding the fourth quantum number, ms, which can have one of two values, +½ and -½.

With the addition of ms, the properties of elements in the Periodic Table could be accounted for. In addition to the atomic number, and the matching of electronic to protonic charge, the reactivities of the elements could be explained in terms of the need to fill the electronic orbitals by sharing electrons, thus explaining the valence properties, and their periodic pattern. Filling of orbital shells, - the building-up (Aufbau) principle

Spin-orbit coupling In addition to the angular momentum and magnetic properties arising from its spin (S), an electron in a molecular orbital has an orbital angular momentum, which also generates a magnetic field (L). The interaction between these two magnetic fields is called spin-orbit coupling. The antiparallel arrangement (bottom, right) has the lower energy, and is therefore favored. The electron magnetic field depends on the total angular momentum, J. In general, both electrons and nucleons spin-couple with other magnets in their environment. This coupling to neighboring spins can be measured in pulsed EPR and NMR applications which look at the kinetics of decay of spin states populated by a pulse of excitation. Analysis of the results provides structural information, - distance and types of atoms close enough to couple.

Some nuclear magnets of importance in biological studies The gyromagnetic ratio (γ) determines the energy at which electromagnetic radiation will flip the spin of a nuclear magnet. A flip occurs when the energy matches (or is in resonance with) the energy of the transition. The energy needed (expressed in terms of frequency) depends on the applied magnetic field (the strength of the magnet), -4.7 Tesla in this case.

What is the difference between the electron and the nucleus as magnets? • The spin quantum numbers for the electron, proton and neutron, all have the same value of ½. This is an intrinsic angular momentum, - every electron, proton and neutron has this property. • Because of the relation between angular momentum and mass, a spin ½ body has a frequency of rotation determined by the mass. Because nuclear particles (nucleons) have masses 1,836  that of the electron, the nucleus spins much more slowly. • Because the magnetic field depends on the rate of rotation, electrons have magnetic fields ~2,000  that of protons, which have higher fields than more massive nuclei. Hence magnetic resonance occurs at much higher energies for EPR than NMR.

2. Wave-Particle Dualism 1.Wave character of particles Diffraction of electron beams X-rays Davisson-Germer experiment Beams of accelerated electrons were directed through thin metal foil and arrival of fast electrons were detected on a photographic plate. The diffraction pattern seen when the electron beam was accelerated to give a wavelength of 0.5 Å gave a pattern similar to that seen when a beam of X-rays of similar wavelength (0.71 Å) was used.

Application: Quantummechanical tunneling The barrier can be defeated by 1) thermal activation (Arrhenius) and 2) tunneling (Gamow) barrier ΔE A particle approaches a potential barrier Δx k=k0·exp(-ΔE/kBT) ΔE T 1.0 Transmittance (Gamow’s expression) • The temperature-dependence of the rates of the two processes are highly different: • Arrhenius: strong temperature-dependence • Tunneling: no temperature-dependence

Estimating the tunneling probability Estimate the relative probabilities that a proton and a deuteron can tunnel through the same barrier of height 1.0 eV and a length 100 pm when their energy is 0.9 eV! The mass of the proton is m = 1.673·10-27 kg, the Planck’s constant is h = 6.626·10-34 J·s the energy difference to overcome by tunneling is ΔE = 0.1 eV and the length of the barrier is Δx = 100 pm. Apply the Gamow-relationship of transmittance via tunneling: Conclusions: 1) the tunneling probability of a proton (in the system specified) is Tp =9.56·10-7 and is much (about 300 times) greater than that of a deuteron: Td =3.07·10-9. 2) The ratio of the tunneling probabilities will be much greater when the barrier is twice as long and the other conditions remain unchanged: Tp/Td = 9·104.

Quantum tunneling effect: Scanning Tunneling Microscope (STM) The electron passes through the gap between the surface of the sample and the tip of the scanning electrode via tunneling. By moving the electrode above the surface, the distance d can be measured and therefore the surface can be mapped.

Application of tunneling in life sciences: Scanning Tunneling Microscope

Electron tunneling in reaction center protein of photosynthetic bacteria LIGHT Upon lowering the temperature from ambient temperature, all electron transfer reaction rates are decreasing and they freeze finally except of the first and fastest reaction (3 ps halftime) which increases by a factor of about 3 at 4 K. This clearly indicates for tunneling pathway instead of a temperature activated reaction of electron transfer between P and HA. Tunneling (e.g. P→BA→HA) k Rate of electron transfer Thermal reactions (e.g. QA→QB) 1/T High temperature Low temperature Arrangement of the Rb. sphaeroidesreaction center cofactors, representation of electron transfer pathways (arrows) and the corresponding time constants. Notations P: bacteriochlorophyll (BChl) dimer, B: monomeric BChl, H: bacteriopheophytine, Q: quinone, Fe: non-heme iron atom, A: photoactive branch, B: photoinactive branch. The circles in the middle of chlorins indicate the Mg atom in BChl.

2. Particle character of wavesThe photoelectric (Hallwachs) effect The limiting wavelengths at the red ends of the photoelectric spectra are indicated in brackets.

The photoelectric effect • When UV light shines on a metal surface, it induces the release of electrons, which can be detected as a current in a circuit such as that on the left. The released electrons are attracted by an applied voltage to an anode, and the resulting current detected, and used to measure the rate of electron release. The characteristics of this effect are as follows: • No electrons are ejected, regardless of intensity, unless the light is sufficiently energetic. In terms of Planck’s equation, they have to have a high enough frequency. The actual value (the work function) depends on the metal. • The kinetic energy of the ejected electrons varies linearly with the frequency of the incident light, but is independent of intensity. • Even at low intensity, electrons are ejected immediately if the frequency is high enough.

According to the classical view, the energy of radiation should be proportional to the amplitude squared. It should therefore be related to intensity, which is in contradiction to the result observed. Einstein suggested a solution to this dilemma, by invoking Planck’s hypothesis. The electron is ejected if it picks up enough energy from collision with a photon. However, the energy of the photon is given by the Planck equation, and so is proportional to frequency, and quantized. From the 1st law of thermodynamics, energy has to be conserved. We can therefore write an equation in which the kinetic energy of the electron is equal to the energy picked up from the photon, minus the energy needed to dislodge the electron (the work function, f): ½mev2 = hn – f This is Einstein’s photoelectric law.

Dual nature of matter – wave and particle properties apply to subatomic particles Einstein’s relation between energy and mass and the de Broglie equation Einstein suggested in the context of special relativity that energy and mass are equivalent, and related through the famous E = mc2. De Broglie realized that, if all matter was quantized, this implied a general relation between the momentum of a particle and its energy as expressed in terms of frequency. By combining the Planck equation, E = hn = hc/l, and the relationship for the momentum of an electromagnetic wave (given by p = mc), p = E/c, we getp = h/λor, rearranging: The de Broglie relationship implies that any particle of mass m moving with velocity v will possess wavelike properties. In view of the value of the Planck constant, the effect will be appreciable for particles of low mass. Conclusion: on atomic scale, the concept of particle and wave melts together, particles take on the characteristics of waves, and waves the characteristics of particles.

Direct application of the de Broglie relation to resolution limit of microscopes The resolution limit of an optical microscope is defined as the shortest distance between two points on a specimen that can still be distinguished by the observer or camera system as separate entities: 0.61·λ/(n·sinα), where n·sinαis the numerical aperture. The wavelength is given by the de Broglie relationship: The resolution limit of the microscope is inversely proportional to the mass and velocity of the particle. The higher is the speed of the particle, the smaller will be the resolution limit. E.g. for electron of v = c/50 in the electronmicroscope, the limit is λ = 0.12 nm (X-ray). The larger is the mass of the particle, the smaller will be the resolution limit. E.g. the (hypothetical) neutronmicroscope would offer about 2000 times smaller resolution limit than the electronmicroscope with the same velocities of the particles because mneutron≈ 2000·melectron The atomic de Broglie microscope is an imaging system which is expected to provide resolution at the nanometer scale using neutral He atoms as probe particles.

3. Unusual spatial distribution and discrete energy levelsExamples:Particles in potential wells • Electron in hyperbolic potential well (spectroscopy of H-type atoms) • Electrons in rectangular potential well (spectroscopy of dyes in the visible range) • Atoms/molecules in parabolic potential well (spectroscopy of vibrations in the infrared spectral range)

1. Electron in hyperbolic potential well(quantization of the H atom) Semiclassical treatment: the forms of the wave functions and the corresponding energy leveles can be anticipated in terms of the de Broglie relation („standing waves”). The Coulomb potential around the nucleus where ε0 is the dielectric constant of the vacuum, e is the elementary charge of the nucleus (proton), r is the distance of the electron from the nucleus. The potential energy of the electron The shape of the function (potential barrier) is hyperbolic. The bound electron should be inside the hyperbolic walls.

The electron is free to move inside the segment but reflected at the boundary. The hyperbolic wall is the constrain that will quantize the energy of the electron. In lack of this condition (i.e. the electron is not bound, it is free), the energy of the electron can take any values, it is continuous. Each wavefunction is - a standing wave, - fits into the segment and - successive functions must possess one more half-wavelength. The permitted wavelengths The average speed of the electron can be estimated by the de Broglie relation Here me is the mass of the electron. The electron has only kinetic energy inside the well thus the kinetic energy of the electron

The total energy of the electron as a function of r The permitted energies are The rigorous solution from the Schrödinger equation (see below) It differs from the semiclassical solution by a factor of π2/4 ≈ 2 only. Conclusions: 1)The energy of the electron in the H atom is quantized, 2) the quantization arises from the boundary conditions that ψmust satisfy and 3) the subsequent energy levels relate as the inverse squares of the neighboring integer numbers: E1 : E2 : E3: ... = 1 : 2-2 : 3-2: ...

The Table on the right shows wave functions, in x, y and z coordinates, for hydrogenic atoms (H, He+, Li2+, etc., - atoms with only 1 electron; Z is the atomic number, giving the charge of the nucleus). The Figures below show the “shapes” (boundary surfaces) of the corresponding orbitals. s The subshells are named s, p, d, f, g, and h for electrons with l = 0, 1, 2, 3, 4 and 5.

Molecular orbitals The energy due to atomic interaction can be calculated as a function of R. The equilibrium bond length is at the minimum. Formation of the bonds in N2. The 2pz orbitals overlap and coalesce into a s molecular orbital; the px and py orbitals form p orbitals perpendicular to each other. Bond wave functions are formed by summing atomic wave functions. Bonds formed by combing antiparallel spins are favored. For H2, the molecular orbital has a lower energy than the atomic orbitals, so is stable. 1s orbitals of H-atoms coalesce to form a sbond of H2, with cylindrical symmetry

Bonds of carbon Carbon has one empty, and two 1-electron filled 2p orbitals, giving four valence electrons. In order to explain the symmetrical chemical behavior of molecules like methane, we have to have four identical orbitals in tetrahedral symmetry. In order to provide these, a mixing of orbitals occurs, to give four sp3 hybrid orbitals (left). This involves promotion of 1 2s electron to the vacant 2p orbital, hence sp3. Figs. from Atkins, The Elements of Physical Chemistry In methane, each sp3 forms a s bond with a 1s atom of H In ethylene (ethene), the double-bond is made up of a s and a p bond.

Take-home message on bonds. • Bonding orbitals can take up quite complicated shapes. Extended p-bonding occurs in molecules like cytochromes, chlorophylls, flavins, nucleic acid bases, tryptophan, etc. The electrons in these extended orbitals roam over the entire coordinated p system (see later the quantumphysical treatment of these systems). • Transitions between energy levels in molecules can occur between orbitals of different type. The change in electron distribution results in an electrical dipole difference between the ground and excited states. When the orbital is asymmetric, as in an extended p system, the excited state can remain in the p-orbital, but will have a different eigenfunction and eigenvalue, and so a different orbital “shape”. This also gives rise to an excited state dipole. • Normally, molecular orbitals are at lowest energy when they are filled by two electrons of antiparallel spin. If an electron is removed (for example, by oxidation), or an extra electron added (for example, by reduction), the lone electron is not spin-coupled, and therefore acts as a magnet. The magnetic effect arises from the unpaired spin, and the magnetic dipole results from the angular momentum of the electron in its orbital.

2. Electrons in rectangular potential well(electronic energy levels) 2π-electrons in conjugated chains Metallic model for the π-electrons: They can move freely along the chain (in the well) but cannot escape (from the well). The shape of the potential well is rectangular and infinitely deep. The double and single bonds between neighboring carbon atoms are alternating. The length of an elementary group is l (≈ 3 Å) the number of elementary groups is N, each group has 2 π-electrons, the whole chain has 2Nπ-electrons and the total length of the chain is L = N·l

Electrons in the box: rectangular potential well There are 2·Nπ-electrons in the well. Because of the bundary conditions, the electrons cannot have arbitrary (continuous set of) energies but well defined discrete values only. We are looking for the possible energy levels using 1) semiclassical physics (based on de Broglie’s standing waves of the electrons) and 2) quantum physics (Schrödinger equations). Semiclassical treatment Only those energy levels are allowed along which standing waves can be created with nodal points at the potential walls. The speeds of the electrons are determined from the de Broglie’s relationship. Replacement of the wavelength in the expression of speed. The total energy consists of kinetic energy term only.

Energies of electrons in the box: rectangular potential well • The energy of the electrons are quantized: • The lowest energy (zero-point energy) is not zero. It comes from the uncertainty principle that requires a particle to possess kinetic energy if it is confined to a finite region. • The energy levels are dense on the bottom and become more diluted upon getting out of the well. The heights of the energy levels are related as the squares of the integer numbers. • According to the Pauli principle, there can be two electrons with antiparallel spins at every level of energy. As there are N groups, the 2·Nπ-electrons occupy the lowest N energy levels. • The separation between adjacent energy levels decreases as the length of the container (L) increases and becomes very small when the container has macroscopic dimensions. In laboratory-sized vessels the energy is not quantized.

The lowest energy band (the most red part) of the absorption spectrum The absorption band with the longest wavelength (smallest energy, the center of the farest red absorption band) If N >>1, then The wavelength of the absorption band is proportional to N.

Discussion The wavelength of the most red absorption band is proportional to N, the length of the conjugated chain. With elongation of the chain, its absorption band shifts toward the longer wavelengths: It shifts to red („red shift”).

molar absorption

3. Particles (atoms and molecules) in parabolic potential well The energy profile of a harmonic oscillator is parabolic. The particle is freely moving inside the parabolic potential well. Semiclassic treatment The total energy of the particle is which is equal to the kinetic energy between the walls. Those energy levels are allowed which can be covered with standing waves Possible velocities of the particle Possible energies of the particle

Energy levels of the harmonic oscillator Uniform ladder of spacing h·ν Solution from the Schrödinger equation: En = h·ν(n + ½), where n = 0, 1, 2, ... • Conclusions: • The energy levels of the oscillator are equidistant: ΔE = h·ν. • The selection rule: Δn = ± 1. • Transitions are allowed between adjacent states only (in first order). Transitions among not neigboring levels are forbidden. • (c) The minimum energy of the oscillator (n = 0) differs from zero: E0 = ½ hν. The zero-point energy is not zero, because of the uncertainty principle: the particle is confined, its position is not completely uncertain, and therefore its momentum, and hence its kinetic energy, cannot be exactly zero. For typical molecular oscillator, the zero-point energy is about E0 = 3·10-20 J = 100 meV The Boltzmann-energy at room temperature is ½ kBT = 25 meV

Quantum Physical Phenomena in Life (and Medical) Sciences