AVERAGE AND STANDARD DEVIATION

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Rosemary Lyons
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1 AVERAGE AND STANDARD DEVIATION Link to: physicspaes home pae. To leave a comment or report an error, please use the auxiliary blo. Post date: 2 Jun 25. Reference: Griffiths, David J. (25), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education - Problems Havin reached the end of the problems in Griffiths s book, I ll rewind and return to chapter to fill in the aps that I left in my rush to et into the meat of quantum mechanics. Probability is at the heart of quantum mechanics, so it s a ood idea to be sure we understand some of the basic concepts. Variables in quantum mechanics come in both discrete and continuous forms, so we ll do a quick review of the averae, variance and standard deviation in these cases. In the discrete case, the averae of a set of values is just the sum of all the values divided by the number of values. To make this precise, suppose that value j occurs N (j) times; then the averae of j is j j jn (j) () j N (j) The averae (also called the mean) is distinct from the median value, which is the value the divides the set into two equal subsets, with one subset containin values less than the median and the other containin values reater than the median. (Obviously this works exactly only if we have an even number of values, but you et the idea.) Havin calculated the averae, we can then calculate the deviation j from that averae for each value j. It turns out that the square of the deviation is more useful, so we have The variance is the mean of these squares: σ 2 ( j) 2 (j j ) 2 (2) ( j) 2 () j 2 2j j + j 2 (4) j 2 j 2 (5) and the standard deviation is the (positive) square root of the variance:

2 AVERAGE AND STANDARD DEVIATION 2 σ ( j) 2 j 2 j 2 (6) The probability of a value j bein chosen at random from the set is P (j) N (j) j N (j) Example. Suppose we have a roup of people with aes as follows: ae j N (j) For this set, we have (7) j (8) j 2 44 (9) j () The median ae is between 22 and 24 (say, 2) since there are 7 people with aes reater than 2 and 7 with aes less than 2. We can now add a column for j: ae j N (j) j Therefore σ 2 We can check equation 5 ( j) () σ 4. (2)

3 AVERAGE AND STANDARD DEVIATION σ 2 j 2 j 2 () () For continuous distributions, we need to use interals instead of sums to calculate the various averaes. To calculate the averae of a function f (x), we need the probability density ρ(x)dx which ives the probability that the value of x lies in the interval (x,x + dx). Given this, we then have f (x) f (x)ρ(x)dx (5) Some special cases of this formula are used to calculate the averae and variance: x x 2 xρ(x) dx (6) x 2 ρ(x)dx (7) A probability density must be normalized so that if f (x) : ρ(x)dx (8) Example 2. Extendin Griffiths s example.. A rock is dropped off a cliff of heiht h. Nelectin air resistance, the position of the rock at time t (usin x as the top of the cliff and x h as the bottom) is x(t) 2 t2 (9) The time taken for the rock to hit the bottom of the cliff is T (2) Now suppose that the position of the rock is randomly sampled a lare number of times durin its fall. If the samplin times are truly random, the probability of a sample bein taken in any time interval is constant and since the probability that a iven sample occurs sometime between t and t T must be, we have ρ(t) T The averae location of the rock over all the samples is then (2)

4 AVERAGE AND STANDARD DEVIATION 4 x T 2 6 h x(t)ρ(t) (22) ( ) 2 t2 T (2) (24) (25) Also The standard deviation is x 2 σ 2 4 h2 5 x 2 (t)ρ(t) (26) ( ) 2 2 t2 T T 4 5 (27) (28) (29) x 2 x 2.298h () To et the probability density for position x instead of time, we have T dx () dx T dx (2) t dx () 2x dx 2 hx (4) ρ(x) 2 hx (5) The probability of a sample bein taken when x is more than σ away from the averae is

5 AVERAGE AND STANDARD DEVIATION 5 ˆ (.298)h ˆ h P σ ρ(x)dx + ρ(x)dx.9 (6) ( +.298)h We could also work this out without first findin ρ(x) by findin the times at which x ( ±.298 ) h from 9 and then interatin ρ(t) over the correspondin time rane. t.298 ) h t ) h ] P σ T [ˆ t + t ) h + (7) (8) (9) ) h (4).9 (4) PINGBACKS Pinback: Gaussian distribution Pinback: Trianular wave function: probabilities

VARIATIONAL PRINCIPLE AND THE HYDROGEN ION: TWO PARAMETERS

VARIATIONAL PRINCIPLE AND THE HYDROGEN ION: TWO PARAMETERS Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog. References: Griffiths, David J. 005), Introduction