The Particle in a Box: Schrödinger Equation, Uncertainty Principle, and Tunneling - Prof. , Study notes of Physical Chemistry

Download The Particle in a Box: Schrödinger Equation, Uncertainty Principle, and Tunneling - Prof. and more Study notes Physical Chemistry in PDF only on Docsity! Lecture 4 The Particle in a Box The Uncertainty Principle The translational partition function Tunneling NC State University Chemistry 431 The particle in a box problem Imagine that a particle is confined to a region of space. The only motion possible is translation. The particle has only kinetic energy. While this problem seems artificial at first glance it works very well to describe translational motion in quantum mechanics. 0 LAllowed Region The solution to the Schrödinger equation with boundary conditions The boundary condition is that the wave function will be zero at x = 0 and at x = L. From this condition we see that B must be zero. This condition does not specify A or k. The second condition is: From this condition we see that kL = nπ. The conditions so far do not say anything about A. Thus, the solution for the bound state is: Note that n is a quantum number! Ψ(0) = Asin(k0) + Bcos(k0) = 0 Ψ(L) = Asin(kL) = 0 or kL = arcsin(0) Ψn(x) = Asin(nπx/L) The Schrödinger equation for a free particle – h 2 2m ∂2Ψ ∂q2 = EΨ The solutions are: Ψ = Aeikq + Be– ikq eikq e-ikq The particle in a box has boundary conditions Ψ(0)= 0 Ψ(L)= 0 L Where is the particle in the box? Since we are using a probability function we do not really know exactly where the particle is. We know that the highest probability occurs for the position L/2. We can guess that this is the average position in the box. However, the more precisely we specify the location of the particle the less information we have about how fast the particle is moving. This is a statement of the famous Uncertainty Principle. ΔxΔp > h/2 Let’s look at the Uncertainty Principle using the particle-in- a-box example. If we know that the particle is in the lowest level then Uncertainty in its position is approximately equal to the width of the probability distribution. The location of a particle in free space is not defined Consider a superposition of a wave with moment hk and h(1.1k) As we add more frequencies we can speak of a bandwidth Δk Δk = 0.1 3 added cosines Δk = 0.2 As the bandwidth increases the position in x-space becomes more defined 5 added cosines The superposition of waves in space leads to the description of a location Δk = 0.7 15 added cosines Gaussian Functions A Gaussian function has the form exp{ -α(x – x0)2 }. The Gaussian indicated is centered about the point x0. The Fourier transform of a Gaussian in x-space is a Gaussian in k-space. Since p = hk we also call this momentum space. The figure shows the inverse relationship. x k Question Which of the following represents the hamiltonian? A. hk B. h 2k2 2m C. – h 2 2m ∂2 ∂x2 D. – ih ∂∂x Question Which of the following represents the hamiltonian? A. hk B. h 2k2 2m C. – h 2 2m ∂2 ∂x2 D. – ih ∂∂x Statistical averaging over translational energy levels Quantum mechanics must agree with classical physics (mechanics) at high temperature or when the average quantum number becomes very large. This is the case for translational energy levels since the spacing of those levels is very small compared to thermal energy, kT. Here, we consider how to average over the energy levels given by the particle-in-a-box solutions. The translational partition function The translational partition function consists of a sum over a very large number of states q = e–(n2–1)βε = e–(n2–1)βεdn 1 ∞ Σ n = 1 ∞ This sum can be expressed as an integral over the states n. The integral can be expressed as a Gaussian. Energies from particle in the box can be used to calculate energy level spacing The difference in energy levels n in the particle- in-the-box solutions has the general form. where ε n gives the energy of a large number of translational energy levels derived from the particle in the box solutions. We can write ε = h2/8mX2 to use a factor in the Boltzmann distribution, so that εn = ε(n2 - 1) εn = h 2 8mX 2 n2 – 1 The translational partition function can be expressed in terms of a thermal wavelength Λ where the thermal wavelength is defined as: q = 2πmkT h2 3/2 V = V Λ3 Λ = 2πmkT h2 Tunneling Tunneling of electrons, protons or other small particles is not possible according to classical mechanics. However, in quantum mechanics a particle can penetrate a barrier even if its kinetic energy,E is less than the potential energy barrier height, V. TunnelingE V Tunneling If the walls of the box are not infinitely high, the The wavefunction of the particle does not decay To zero. Instead it decreases exponentially. Ultimately it has some probability for tunneling Through a barrier even when E < V. – h 2 2m ∂2 ∂x2 Ψ + VΨ = EΨ TunnelingE V Boundary conditions The boundaries are at x = 0 and x = L. Thus, the conditions are: The first derivatives are: There are four equations and six unknowns. If we assume that the particle is coming from the left then we can set B = 0. If we calculate the transmission coefficient then we need the ratio A’/A. A + B = C + D at x = 0 A′eikL + B′e–ikL = CeκL + De–κL at x = L ikA – ikB = κC – κD at x = 0 ikA′eikL – ikB′e–ikL = κCeκL – κDe–κL at x = L Transmission coefficient With these constraints we can solve for the transmission coefficient, T. where ε = E/V. The tunneling phenomenon is important in electron transfer theory. Electron transfer is a key aspect of energy transduction in biology (e.g. photosynthesis and respiration, among others). Electron tunneling is also the effect used in the scanning tunneling microscope (STM). T = 1 + e κL – e–κL 16ε(1 – ε) – 1