Chapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II)

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1 Chapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II) The Particle in the Box and the Real World, Phys. Chem. nd Ed. T. Engel, P. Reid (Ch.16)

2 Objectives Importance of the concept for particle in the box Understanding the tunneling of quantum mechanical particles

3 Outline 1. The Particle in the Finite Depth Box. Pi Electrons in Conjugated Molecules Can Be Treated as Moving Freely in a Box 3. Tunneling through a Barrier 4. The Scanning Tunneling Microscope 5. Tunneling in Chemical Reactions

4 8.5 The Particle in the Finite Depth Box For a box to be more realistic, we let the box to have a finite depth. The potential is defined by V V ( x) = 0, for a / > x > a / ( x) = V, for x a /, x / 0 a Outside the box, ( ) ( ) ψ x ψ x where = Ae = A'e k κx κx = + Be + κx + B'e m V + κx ( E) h 0 for x a/ for - x a/,

5 8.9 Tunneling through a Barrier Consider a particle with energy E confined to a very large box. A barrier of height V 0 separates two regions in which E < V 0. The particle can escape the barrier and go over the barrier, called tunneling.

6 8.9 Tunneling through a Barrier To investigate tunneling, finite depth box is modified by having a finite thickness. The potential is now V V V ( x) = 0, for x < 0 ( x) = V0, for 0 ( x) = 0, for x > a x a where a = barrier width

Physical Chemistry Fundamentals: Figure 8.8 p.97 Fig. 8.8 A particle incident on a barrier from the left has an oscillating wave function, but inside the barrier there are no oscillations (for E < V).

7 Physical Chemistry Fundamentals: Figure 8.8 p.97 Fig. 8.8 A particle incident on a barrier from the left has an oscillating wave function, but inside the barrier there are no oscillations (for E < V). If the barrier is not too thick, the wavefunction is nonzero at its opposite face, and so oscillates begin again there. (Only the real component of the wavefunction is shown.)

8 Tunnelling The particle might be found on the outside of a container even though according to classical mechanics it has insufficient energy to escape. Such leakage by penetration through a classically forbidden region is called tunnelling. Schrödinger eqn -> the probability of tunnelling of a particle of mass m incident on a finite barrier from the left. On the left of the barrier (for x < 0) the wavefunctions of a particle with V = 0,

9 Tunnelling Schrödinger equation for the region representing the barrier (for 0 x L), where the potential energy is the constant V (E<V) To the right of the barrier (x > L), where V = 0 again, the wavefunctions are

10 Tunnelling Key points Tunnelling is the penetration into or through classically forbidden regions. The transmission probability decreases exponentially with the thickness of the barrier and with the squareroot of the mass of the particle.

11 Physical Chemistry Fundamentals: Figure 8.9 Fig When a particle is incident on a barrier from the left, the wavefunction consists of a wave representing linear momentum to the right, a reflected component representing momentum to the left, a varying but not oscillating component inside the barrier, and a (weak) wave representing motion to the right on the far side of the barrier.

12 The acceptable wavefunctions must be continuous at the edges of the barrier (at x = 0 and x = L, e 0 = 1):

Physical Chemistry Fundamentals: Figure 8.10 The slopes of wavefunctions (their first derivatives) must also be continuous at the edges of the barrier (at x = 0 and x = L) Fig. 8.10 The wavefunction and its slope must be continuous at the edges of the barrier.

13 Physical Chemistry Fundamentals: Figure 8.10 The slopes of wavefunctions (their first derivatives) must also be continuous at the edges of the barrier (at x = 0 and x = L) Fig The wavefunction and its slope must be continuous at the edges of the barrier. The conditions for continuity enable us to connect the wavefunctions in the three zones and hence to obtain relations between the coefficients that appear in the solutions of the Schrödinger equation.

14 Tunnelling 4 eqns for 6 unknown coefficients. If the particles are shot towards the barrier from the left, there can be no particles travelling to the left on the right of the barrier. Setting B = 0 removes 1 more unknown. We cannot set B = 0 because some particles may be reflected back from the barrier toward negative x. The probability that a particle is travelling towards positive x (to the right) on the left of the barrier is proportional to A, and the probability that it is travelling to the right on the right of the barrier is A. The ratio of these two probabilities is called the transmission probability, T. (ε=e/v)

15 Physical Chemistry Fundamentals: Figure 8.11 Fig The transition probabilities for passage through a barrier. The horizontal axis is the energy of the incident particle expressed as a multiple of the barrier height. The curves are labelled with the value of L(mV) 1/ /ħ. The graph on the left is for E < V and that on the right for E > V. Note that T > 0 for E < V whereas classically T would be zero. However, T < 1 for E > V, whereas classically T would be 1.

16 Physical Chemistry Fundamentals: Figure 8.1 Fig. 8.1 The wavefunction of a heavy particle decays more rapidly inside a barrier than that of a light particle. Consequently, a light particle has a greater probability of tunnelling through the barrier.

17 Tunnelling For high, wide barriers (in the sense that κl» 1), eqn 8.19a simplifies to (8.19b) The transmission probability decreases exponentially with the thickness of the barrier and with m 1/. Particles of low mass are more able to tunnel through barriers than heavy ones (Fig. 8.1). Tunnelling is very important for electrons and muons, and moderately important for protons; for heavier particles it is less important. A number of effects in chemistry (e.g., the isotope-dependence of some reaction rates) depend on the ability of the proton to tunnel more readily than the deuteron. The very rapid equilibration of proton transfer reactions is also a manifestation of the ability of protons to tunnel through barriers and transfer quickly from an acid to a base. Tunnelling of protons between acidic and basic groups is also an important feature of the mechanism of some enzyme-catalysed reactions. Electron tunnelling is one of the factors that determine the rates of electron transfer reactions at electrodes and in biological systems.

18 Physical Chemistry Fundamentals: Figure 8.13 The wavefunction penetrates into the walls, where it decays exponentially towards zero, and oscillates within the well. The wavefunctions and their slopes are continuous at the edges of the potential. Fig A potential well with a finite depth.

19 Physical Chemistry Fundamentals: Figure 8.14 Finite number of bound states - always at least one bound state. N the number of levels: The deeper and wider the well, the greater the number of bound states. Fig The lowest two boundstate wavefunctions for a particle in the well shown in Fig

20 Physical Chemistry Fundamentals: Figure 8.15 Fig A scanning tunnelling microscope makes use of the current of electrons that tunnel between the surface and the tip. That current is very sensitive to the distance of the tip above the surface.

21 Physical Chemistry Fundamentals: Figure 8.16 Fig An STM image of caesium atoms on a gallium arsenide surface.

22 The Scanning Tunneling Microscope Tunneling through a Barrier Scanning Tunneling Microscope (STM) allows the imaging of solid surfaces with atomic resolution with a surprisingly minimal mechanical complexity. The STM is used to study the phenomena at near atomic resolution.

23 8.10 The Scanning Tunneling Microscope Scanning Tunneling Microscope (STM)

24 Example As was found for the finite depth well, the wave function amplitude decays in the barrier according to ( ) [ ( ) ] ψ x = Aexp m V0 E / h x. This result will be used to calculate the sensitivity of the scanning tunneling microscope. Assume that the tunneling current through a barrier of width a is proportional A exp [ ( ) ] m V E / h a 0

25 Example a. If V E is 4.50 ev, how much larger would the 0 current be for a barrier width of 0.0 nm than for 0.30 nm? b. A friend suggests to you that a proton tunneling microscope would be equally effective as an electron tunneling microscope. For a 0.0-nm barrier width, by what factor is the tunneling current changed if protons are used instead of electrons?

26 Solution a. Putting the numbers into the formula given, we obtain I I 10 ( a =.0 10 m) 10 ( a = m) = exp = exp = 8.78 m V ( E ) ( ) h ( )( 4.50)( ) 34 ( ) 10 ( ) Even a small distance change results in a substantial change in the tunneling current.

27 Solution b. We find that the tunneling current for protons is appreciably smaller than that for electrons. I ( proton) ( electron) I exp = exp = exp = exp = m m proton 0 electron ( V E) ( V E) ( V ) 0 E ( m m ) h h h 0 19 ( 4.50)( ) 34 ( ) ( ) (.0 10 ) This result does not make the proton tunneling microscope look very promising. a a proton electron a

28 Tunneling Most chemical reactions proceed faster as the temperature of the reaction mixture is increased. This is due to energy barrier which must be overcome in order to transform reactants into products. This barrier is referred to as the activation energy for the reaction.

29 Tunneling

Lecture 5. Potentials

Lecture 5 Potentials 51 52 LECTURE 5. POTENTIALS 5.1 Potentials In this lecture we will solve Schrödinger s equation for some simple one-dimensional potentials, and discuss the physical interpretation