Oscillographic Measurements of Probability Distributions

J. H. E. Griffiths

Abstract A method of determining directly the mean lives of excited atoms was described in detail in a previous paper. This method employs a Kerr cell operated by a high frequency oscillator which is also used to run a Geissler tube. With the electrical arrangement used, a minimum of light should pass through the Kerr cell system if light were emitted instantaneously. Owing to the finite time that the atoms remain in the excited state, there is a lag between the current in the discharge tube and the emission of light. By increasing the distance that the light travels before reaching the cell a point may be found where a maximum of light passes the cell, from which the lag may be determined. This is directly connected with the mean life of the state. In the previous paper results for neon were given and the present paper deals with the application of this method to hydrogen. Fig. 1 shows the electrical arrangement used. This is essentially the same as that used for neon, except that a cathode ray oscillograph is used to determine the phase difference between the potential on the Kerr cell and the current in the discharge tube. Thus the deflector plates Px give deflections proportional to the voltage on the Kerr cell K. C. while the plates Py give deflections proportional to the voltage on C2 and therefore in phase quadrature with the current through the discharge tube. L1C1 is a tuned circuit which picks up energy from the oscillator and by tuning this, the oscillograph figure may be made a straight line. When this is so the current through the discharge tube is 90° out of phase with the voltage on the Kerr cell. L2C2 forms a series tuned circuit of low impedance, and the variable condenser C7 gives a means of making the two deflections equal. The oscillograph can be set for linearity within an accuracy of ½° which is of the same order as the accuracy of the experiment. A correction has to be applied for the time that the electrons take to travel between the plates of the oscillograph. Thus when the same oscillating voltage is applied to the two pairs of plates, the oscillograph figure is not a straight line, but shows an apparent phase difference, owing to the fact that one pair of plates acts on the electron beam sooner than the other pair. With the oscillograph used, this time was 1⋅5 . 10-9 secs, or at a wave-length of 60 metres a phase angle of 2⋅7°. This must be added to the observed lag.

Lars M. Johansen

Patrick P. Hofer

We develop a general framework to investigate fluctuations of non-commuting observables. To this end, we consider the Keldysh quasi-probability distribution (KQPD). This distribution provides a measurement-independent description of the observables of interest and their time-evolution. Nevertheless, positive probability distributions for measurement outcomes can be obtained from the KQPD by taking into account the effect of measurement back-action and imprecision. Negativity in the KQPD can be linked to an interference effect and acts as an indicator for non-classical behavior. Notable examples of the KQPD are the Wigner function and the full counting statistics, both of which have been used extensively to describe systems in the absence as well as in the presence of a measurement apparatus. Here we discuss the KQPD and its moments in detail and connect it to various time-dependent problems including weak values, fluctuating work, and Leggett-Garg inequalities. Our results are illustrated using the simple example of two subsequent, non-commuting spin measurements.

Emily Marshman, Chandralekha Singh

A solid grasp of the probability distributions for measuring physical observables is central to connecting the quantum formalism to measurements. However, students often struggle with the probability distributions of measurement outcomes for an observable and have difficulty expressing this concept in different representations. Here we first describe the difficulties that upper-level undergraduate and PhD students have with the probability distributions for measuring physical observables in quantum mechanics. We then discuss how student difficulties found in written surveys and individual interviews were used as a guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop a good grasp of the probability distributions of measurement outcomes for physical observables. The QuILT strives to help students become proficient in expressing the probability distributions for the measurement of physical observables in Dirac notation and in the position representation and be able to convert from Dirac notation to position representation and vice versa. We describe the development and evaluation of the QuILT and findings about the effectiveness of the QuILT from in-class evaluations.

Jan Naudts