- develop a theory that describes the wave properties of particles correctly

Transcription

1 Quantum Mechanics Bohr's model: BUT: In : by 1930s: - one of the first ones to use idea of matter waves to solve a problem - gives good explanation of spectrum of single electron atoms, like hydrogen - does not explain even simple many electron systems, like Helium - does not explain intensity of spectral lines - does not help to explain structure of atoms, their binding to other atoms, or any physical of chemical properties of interacting atoms - another more general approach to explain properties of microscopic system such as atoms is developed by Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac and others - it is called quantum mechanics - a vast number of experiments in physics and chemistry is successfully explained using this new theory - up to now the theory of quantum mechanics has been proven to be correct by many experiments - quantum mechanics is a very successful theory phys4.4 Page 1 Goal: steps: - develop a theory that describes the wave properties of particles correctly - use a complex valued wave function ψ to describe the particle - the modulus squared of ψ evaluated at a specific coordinate and time is proportional to the probability density P to find a particle at that coordinate and at that time - ψ is the complex conjugate wave function to ψ properties of ψ: - the wave function is normalized - if ψ describes the particle and ψ 2 its probability density then the total probability of finding the particle somewhere has to be unity (=1) for the wave function to be a valid description of the particle phys4.4 Page 2

2 properties of ψ: - single valued and continuous - single valued and continuous derivatives probability - to find a particle in a region of space (for simplicity in one dimension): Schrödinger equation: exercise: - fundamental equation of quantum mechanics - the wave equation for matter waves - is postulated and cannot be derived (like other fundamental equations, laws of thermodynamics, Newton's equation of motion, ) - work out some examples of wave equations to remember some of their properties and get some familiarity working with these phys4.4 Page 3 time-dependent Schrödinger equation: wave function: - for a free particle described by a plane wave along the x- direction with velocity v = λ ν - express ν and λ in terms of energy and momentum - thus - with the reduced Planck constant - This wave function describes a free particle. How to find the wave function describing a particle in the presence of external constraints, e.g. a potential resulting in some force acting on the particle? goal: - Find a fundamental differential equation the solution to which is the wave function describing the properties of the particle. phys4.4 Page 4

3 Schrödinger equation: - a strict derivation from first principles is not possible - let's develop some intuition trying to set up a wave equation for ψ by differentiating it with respect to x - and with respect to t - for v < c the total energy of a particle is given by - multiplying with ψ phys4.4 Page 5 - substitute expressions for E and p - time-dependent Schrödinger equation (1D) - in three dimensions (3D) - the potential function U contains all the external forces acting on the particle - solving the Schrödinger equation for ψ the probability density for the particle can be found for any coordinate and time - again: Schrödinger's equation is a postulate, there is no formal justification in the considerations presented above - but the physical world as probed in exquisite detail in experiments is very well described by this equation phys4.4 Page 6

4 Linearity: Superposition: - the Schrödinger equation is linear in the wave function ψ - it contains no terms that involve higher powers of ψ or it's derivatives - as a result if ψ 1 and ψ 2 are solutions of a S.E. the superposition of both is also a solution - a 1 and a 2 are complex coefficients - the superposition principle applies to matter waves in similar ways as to any other waves - the superposition principle gives rise to interference effects in quantum mechanics single solution terms two-solution terms phys4.4 Page 7 Expectation values: position expectation values: - all information about the particle is contained in the wave function ψ - we know already how to calculate the probability to find a particle at a certain coordinate and time - to extract other experimentally accessible quantities expectation values are calculated - classical center of mass for a collection of N particles with N i particles at position x i - for a single particle with probability p i at position x i - in the continuous limit phys4.4 Page 8

5 general: - expectation values of quantity G(x) (that is not an operator!) - What about the expectation values of the momentum p and the energy E? - differentiate the free particle wave function ψ with respect to x and t - correspondence phys4.4 Page 9 Operators: - momentum operator - energy operator - total energy operator: - multiply with ψ to get back to the Schrödinger equation - expectation values of an operator O - for example for the momentum phys4.4 Page 10

6 example: - expectation value of the momentum p of a free particle time independent (steady state) Schrödinger equation - in many problems the potential U in the Schrödinger equation does not depend on time explicitly - in this case the Schrödinger equation can be simplified - note for the free particle wave function - for all time-independent problems the wave function can be separated into these two parts phys4.4 Page 11 time-independent (steady-state) Schrödinger equation: - in three dimensions - this steady-state Schrödinger equation may have one or more solutions ψ i - the energy E i corresponding to the different solutions ψ i naturally explain the energy quantization observed in any stable timeindependent system phys4.4 Page 12

7 Eigenfunctions and eigenvalues of operators: - consider an operator O measurement postulate: - if this equation holds ψ n is an eigenfunction of the operator O with eigenvalue o n - the equation above is also called an eigenvalue equation - if a physical observable O corresponding to the operator O is measured the only possible measurement outcomes are any of the o n phys4.4 Page 13 Hamilton operator: - the total energy operator E of a system can be written in terms of the momentum and potential energy operators p, U in - with the Hamilton operator H - the Schrödinger equation thus is a eingenvalue equation for the Hamiltonian H operators: - the E n are the eigenenergies of a system described by the Hamiltonian H, they are corresponding to the quantized energies of the system position momentum potential energy kinetic energy total energy phys4.4 Page 14

8 Particle in a box: - a single particle trapped in a box with hard walls - one of the simplest possible examples of solving a Schrödinger equation - let's treat the box as one-dimensional (1D) - the particle is confined along the x-direction at x = 0 and at x = L - in the region 0 < x < L the potential is U = 0, elsewhere U = - plot of potential U phys4.4 Page 15 Particle in a box: - task: find the wave function ψ describing the particle in the box by solving the Schrödinger equation - 2 nd order ordinary differential equation for ψ - general solution - boundary condition ψ = 0 for x < 0 and x > L phys4.4 Page 16

9 boundary conditions: - continued - resulting allowed particle energies E n - these are the allowed energy levels of a particle in a 1D box wave function: - the corresponding (not yet normalized) wave functions - normalization condition normalized wave function: phys4.4 Page 17 plots for different n: - wave functions ψ n - probability densities ψ n 2 note: - the probability densities vary a lot with n - n = 1: largest probability in the center (x = L/2) - n = 2: zero probability in the center - classically probability is expected to be the same everywhere phys4.4 Page 18

10 expectation values: - position <x> - momentum <p> - the expectation value of the momentum vanishes! Why is that? - expectation for momentum from energy E = p 2 /2m - average momentum phys4.4 Page 19 momentum eigenfunction: - momentum eigenvalue equation - energy eigenfunctions are not momentum eigenfunctions - momentum eigenfunctions momentum eigenvalues phys4.4 Page 20

11 Finite Potential Well: - in a 1D potential well where U = 0 for 0 < x < L and U = U 0 < the particle can penetrate into the wall - all natural potential wells have finite depth - consider a particle with energy E < U 0 less than the depth of the potential - three regions in the problem - region II: U = 0 - region I and III: U 0 > E phys4.4 Page 21 general solution: - in regions I and III - otherwise ψ would diverge - in region II - with boundary conditions continuity of wave function continuity of momentum phys4.4 Page 22

12 finite potential well wave functions: probability densities: some notes: - the particle is trapped forever - how does the particles ever get into the potential well? - how does it leave? - physical examples of a particle in a box phys4.4 Page 23 phys4.4 Page 24

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