Probability One of the most useful and intriguing aspects of quantum mechanics is the Heisenberg Uncertainty Principle. Before I get to it however, we need some initial comments on probability. Let s first look at probability with discrete variables. For instance, the probability of a coin toss giving heads. Everyone knows that this is 50%. In order to determine this experimentally however, I need to toss the coin many times, or alternatively, I can toss many identical coins once. Let us begin with this example to create a pattern that is based on intuition. Let s make a plot of the value of the coin toss (heads 1, tails -1) and histogram what happens after many (200) tosses (for simplicity of my arguments, I happen to make equal heads and tails tosses): 100-1 1 Now, let s ask some questions about this distribution of events: 1) What is the probability of getting tails (-1)? N( 1) 100 1 A) We can calculate the probability by: P ( 1 ) N( 1) + N( 1) 100 + 100 2 2) What is the probability of getting heads (1)? A) OK, I won t write down the equation, P(1) ½. Notice that the sum of the probabilities is given by: N( 1) N( 1) 100 + 100 P ( 1 ) + P( 1) + 1 N( 1) + N( 1) N( 1) + N( 1) 100 + 100 the probability of getting either heads or tails must be 1 (excluding the remote possibility of getting it to land on its edge). 3) What is the most likely value? A) Each is equally likely. 4) What is the median value? A) The median value is defined to be the number where the probability of getting a larger value is the same as the probability of getting a smaller value. In our case here, the median is zero. 5) What is the average value?
A) I sometimes get median and average confused. Often times, distributions have the same median and average values, but this is not always the case. In our present case, we calculate the average by: 1 100 + 1 100 Average 0, 200 and it is the same as the median. It is important to note here that we will never actually measure the average value. As a more instructive example, I want to take some text straight out of Griffiths, since I don t think that I can summarize it as well as he has done in his excellent text:
OK, so what should really be taken away from this? First, equation 1.9 is very important to be comfortable with. The average value (or expectation value as it is used in QM, but be careful with that word!) of any function f is just the sum of f(j) times the probability of j or P(j). A good understanding of the variance and the standard deviation is also very important, and will lead us soon to a discussion of the uncertainty principle. First, let s build on what we have learned with discrete variables, and move to continuous variables. To build on the previous example, what happens when we ask what the probability is of someone having the exact same birthday as me, down to the microsecond? It would be very, very small, in fact perhaps zero. For continuous variables, it only makes sense to talk about the probability of something happening in a particular interval, say the probability that someone has a birthday the same as mine within one day. This then would be the probability per unit time, or a probability density ρ(t). So, the probability of finding someone with the same birthday in an interval from t to t + dt would be given by ρ(t)dt. For a finite interval, say from July 1 to July 31, one would sum up all the probabilities for each (infinitesimal) time interval, i.e., integrate from the beginning of the July 31 interval to the end: P( Birthday is in July) ρ() t July 1 dt
where ρ(t) is the probability density for someone to have a birthday in a particular time interval. You may catch me saying that ρ(t) is the probability that something will happen. Just remember what I mean, and not what I say. With this definition, we can now delineate several rules that apply to probabilities: The first is just a statement that the total probability must be one. In our birthday example, the probability that someone has a birthday between Jan. 1 and Dec. 31 is one. The probability that a coin toss will result in a heads or a tails is one, etc. We call this normalization of the probability density. The second is just a definition of the average value over the complete interval. The third is a generalization of the second, meaning that the average value of any function of the variable is just given in the same way as the average value, i.e, the function times the probability density integrated over the complete interval. The fourth equation is just a restatement of the definition of the variance. You will use these over, and over again throughout this semester, so become familiar and comfortable with them. NOTE: Many texts use x for σ x! In fact, I was taught that way and Liboff uses that notation. The formalism of Griffiths above is, I believe, a more standard mathematical notation. I will try to use σ throughout this semester, but be aware of the possibility of confusion. Homework: Consider a Gaussian distribution (quite common): ρ x ( ) Ae λ x a ( ) 2 where A, a and λ are positive real constants. a) Find the constant A by using the normalization condition. b) Find <x>, <x 2 > and σ. c) Sketch the graph of ρ(x).