Consider the time evolution of and . This can be visualized by plotting the point ( , ) in the - plane. This plane is generally known as phase-space. In general, as time progresses, the point ( , ) will trace out some very complicated pattern in phase-space. Suppose that we divide phase-space into rectangular cells of uniform dimensions and . Here, is the intrinsic error in the position measurement, and the intrinsic error in the momentum measurement. The area of each cell is
(3.1) |
Let us now consider a single spinless particle moving in three dimensions. In order to specify the state of the system, we now need to know three - pairs: that is, - , - , and - . Incidentally, the number of - pairs needed to specify the state of the system is usually called the number of degrees of freedom of the system. (See Section B.1.) Thus, a single particle moving in one dimension constitutes a one degree of freedom system, whereas a single particle moving in three dimensions constitutes a three degree of freedom system.
Consider the time evolution of and , where , et cetera. This can be visualized by plotting the point ( , ) in the six-dimensional - phase-space. Suppose that we divide the - plane into rectangular cells of uniform dimensions and , and do likewise for the - and - planes. Here, and are again the intrinsic errors in our measurements of position and momentum, respectively. This is equivalent to dividing phase-space up into regular six-dimensional cells of volume . The coordinates and can now be conveniently specified by indicating the cell in phase-space into which they plot at any given time. Again, this procedure automatically ensures that we do not attempt to specify and to an accuracy greater than our experimental error.
Finally, let us consider a system consisting of spinless particles moving classically in three dimensions. In order to specify the state of the system, we need to specify a large number of - pairs. The requisite number is simply the number of degrees of freedom, . For the present case, , because each particle needs three - pairs. Thus, phase-space (i.e., the space of all the - pairs) now possesses dimensions. Consider a particular pair of conjugate (see Section B.4) phase-space coordinates, and . As before, we divide the - plane into rectangular cells of uniform dimensions and . This is equivalent to dividing phase-space into regular dimensional cells of volume . The state of the system is specified by indicating which cell it occupies in phase-space at any given time.
In principle, we can specify the state of the system to arbitrary accuracy, by taking the limit . In reality, we know from Heisenberg's uncertainty principle (see Section C.8) that it is impossible to simultaneously measure a coordinate, , and its conjugate momentum, , to greater accuracy than . Here, is Planck's constant divided by . This implies that
(3.2) |
In quantum mechanics, we can specify the state of the system by giving its wavefunction at time ,
(3.3) |