How does it work? QM describes the microscopic world in a way analogous to how classical mechanics (CM) describes the macroscopic world.

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1 Today we use Quantum Mechanics (QM) on a regular basis: Chemical bonds NMR spectroscopy The laser (blue beam in Blue-ray player; red beam in a DVD player for example) The fingerprint /signature of a molecule or atom in any spectrum-due to quantization; that is, if any energy were allowed there would be no fingerprint These days many materials have important structural features at the atomic/molecular scales; understanding their properties require QM How does it work? QM describes the microscopic world in a way analogous to how classical mechanics (CM) describes the macroscopic world.

2 Say we have a particle of mass m moving along the x-axis under a force F(c,t) m F(x,t) In CM to learn about the particle we describe it s position at any given time, t by x(t). One we know that, we know its momentum and kinetic energy To get x(t) we use Newton s second law: x V = potential energy. Knowing x(0) and V(0) allows x(t) to be found.

3 QM approaches the problem differently. Here the function are looking to describe the system is the wavefunction Ψ(x,t) of the particle. To get that we solve the Schrodinger equation: Here: and We will work a lot with this equation but before we do, we ll deal with some mathematical background that will prepare us to come to the Schrodinger equation successively at various levels of sophistication.

4 Review of Probability and Statistics Due to the probabilistic nature of quantum mechanics, it is useful to review some basic concepts of probability and statistics. Consider an experiment (such as rolling of a dice) which has n possible outcomes, each with a probability P j. The probability of outcome j occurring is given by P j With each experiment (or roll of the dice) it is certain a result will occur. In other words there is a 100% probability of an event occurring. Thus, the sum of all probabilities is unity since one of the n outcomes will occur. Consider a 4-sided die. The probabilities are given in the table: outcome probability total:

5 Now let us associate a value x j with each outcome. For example, say with our 4-sided die we have numbers 1, 3, 7 and 10 on the sides, instead of 1, 2, 3 and 4. outcome, j result, x j Probability, P j The average or mean of the result x is given the symbol <x> and is given by: In the specific example of our 4-side die above we have: 5

6 The variance, s 2, indicates the extent to which the set of outcomes or observations cluster around the mean value. The variance is equal to the average of the squared deviations from the mean: The standard deviation, s, is the positive root of the variance: 6

7 probability probability Discrete and Continuous Probabilities With the case of the dice, these were discrete probability distributions: In this case the probability distribution function is discrete outcome P j In this course we will more often have to deal with continuous probability distributions. For example the mass of water melons in a large batch will be given by a continuous probability since the mass of the melons can be any value greater than zero. Px ( ) Such a probability distribution can be described by a continuous function P(x) outcome (mass in pounds) 7

8 probability In our example, the probability of finding a water melon between mass a and mass b is given by the integral: a b Px ( ) outcome (mass in pounds) As with the discrete probabilities, the sum of all probabilities must sum to unity or 100%. Thus, our continuous distribution function must satisfy the following equation: In the case the function describing the mass of water melons we have: 8

9 In analogy to the discrete distributions, the average outcome given by a continuous probability distribution is: Example A very common probability distribution is a Gaussian distribution. Px ( ) x=0 9

10 Let us show that the probability distribution function sums to unity. From a table of integrals this integral is: What integral would we have to evaluate to determine the average outcome and what should the average be? answer: The average should be zero based on the symmetry of the probability around x=zero 10

11 Standard Deviation The standard deviation, σ, gives a measure of how spread out the results are from the mean. Px ( ) For a continuous distribution the standard deviation defined in terms of the variance, σ 2, in analogy to the discrete definition. Px ( ) s x=0 11

12 Complex numbers Complex numbers can be written as: a In polar coordinates b Complex plane Engel: page 455.

13 Euler relations Complex conjugate Addition Multiplication

14 In class: Prove De Moivre s theorem: Division of complex numbers

15 Example: What is the Cartesian form of: z= 2 5i 5 i

16 Differential Calculus Derivatives are slopes of the function at the point of interest. df x = h 0 lim f x+h f x h Rules d ax n =anxn 1 d a sin x =a cos x d a cos x = a sin x d ae bx =abe bx d f+g = df + dg d f x g x =f x dg x df x +g x Product rule df u x = df u du du Chain rule Can do higher order derivatives too.

17 Partial derivatives: for functions that depend on several variables. Example: f x,y =x 2 e 2yx Evaluate: 2 f x,y y x x f f +y x y

18 Integrals -definite and indefinite if the integration is over a domain the result is a function of that domain, not the integration variable Geometric Interpretation S = area under the curve = Indefinite integrals have an unknown constant in the result because the derivative of a constant is 0.

19 Integration by parts: Useful particularly for integrals where a power of the variable is involved. Therefore integral = do integration by parts again

20 Review on waves Classical waves:

21 when waves collide principle of superposition For the diffraction experiment δ = phase difference