*Fig. 4: Semi-log plot of the histogram showing the distribution of 5000 measurements of timeintervals between a photon event and its first successor; horizontal axis in microseconds*

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*Fig. 1: Amplified pulses from the PMT and delayed logic pulses from the Monitor Output, indicating the arrival of photons and signals above threshold. The 'scope was set on persistence mode to display multiple triggered events. *

The histogram is not Gaussian, but exponential, in character. The mean interval implied by this distribution is 0.910 ms, to be compared with the value 0.921 ms expected from the average count rate from Fig. 2. But the histogram does not have a peak at that value; in fact, the shortest time intervals are the most probable.

Note too, that the degree to which the distribution matches the expected exponential is a test of the random-event hypothesis: any correlation between photon-detection events, or any periodicity in their arrival, would show up directly in this plot.

There are many other sources of electronic events, some highly (but not perfectly) regular, others nearly (or wholly) random, and all of them susceptible to statistical study. You might think of a not-quite-perfect clock, or a Geiger counter, as sources of such events. The pre-requisite for such statistical studies is a large sample of data, and we’ve now made that as easy, and straightforward, as possible to obtain. We’ve paid attention to transparency of operation of our new counter/timer, ensuring (for example) that data can be taken by hand for a first mode of understanding of what’s going on. We feel sure that students would do best to take a first (small) data set manually, so that they understand the process of abstraction and aggregation that lies between a list of data and the histogram derived from it.

Returning to the __ Two-Slit experiment__, we want students to be able to see that the photon events they’re detecting, whether at an interference maximum or a minimum, still display all the statistical properties expected for events that are random and uncorrelated, but nevertheless occurring at a well-defined mean rate. And this invites an even deeper look into the essential quantum mystery.

Finally, we expect that this counter/timer will have uses far beyond the Two-Slit experiment. We invite you to come up with some of them.

*Fig. 2: Scatter plot of number of photon events detected in 1-second gate times, plottedas a function of observation number, for 1200 successive observations*

TeachSpin’s ‘__ Two-Slit Interference, One Photon at a Time’ __has become a classic demonstration of the ‘essential

quantum paradox’ and an experiment in single-photon detection. Now, we’ve improved its electronics with a new

Pulse Counter/Interval Timer unit which makes possible a whole new set of investigations in the *statistics of *

*random-event processes.***Our NEW Pulse Counter/Interval Timer unit will:**

* discriminate pulse events from a noise background, for pulses of amplitude 10 mV to > 1 Volts;

* correctly count ‘events per unit time’, with 0.1, 1.0, or 10-second counting intervals;

* alternatively, measure the time interval between a pulse and its successor pulse, to 1- µs resolution;

* and write data files of unlimited length (via HyperTerminal, in a host computer) of successive counts per unit time, or intervals between successive counts.

An electronic discriminator, designed to detect even low-level analog pulses standing up above a noise background, is built into the counter/timer unit. The discriminator’s threshold level can be adjusted continuously and linearly over a range of about 10 mV to several Volts. A monitor output gives a view of exactly what input pulses are meeting the discriminator’s threshold criterion. Figure 1 shows the discriminator in action.

Back to the plot in Fig. 2: it shows counts arriving at an average rate of about 10 /s, which means that counts are arriving at average time interval of 10 s or 1 ms.

But, if that time interval between a pulse and its first successor is measured repeatedly, what individual values will result? And what distribution of values will result from repeated series of such measurements? Do you know what histogram those values will have?

Our new Counter/Timer can be configured to measure such a time interval, and to do so repeatedly, and write the results to a file, easily giving a sample of (say) 5000 time-interval measurements. In such a sample, we found (for example) nearly 1000 occurrences of time intervals in the 1-200 µs range. The histogram of all 5000 measured intervals is shown in Fig. 4.

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The data display statistical fluctuations about the mean, of 1086 counts/second and students can process the data files from our new counter to study these fluctuations. Questions they should explore include:

* What is the standard deviation of the count rate?

* How does that result depend on the (variable) average count rate?

* What does the chi-squared test have to say about the data in Fig. 2?

* What is the histogram of the data in Fig. 2? (See Fig. 3.) Is this distribution Gaussian, and does it have the width expected from Poisson statistics?

*Fig. 3: Histogram of the data of Fig. 2, showing also the Gaussian distribution expectedfrom the central-limit theorem distribution expected from the central-limit theorem*