Wave packet uncertainty relation

Another Heisenberg uncertainty relation exists for the energy E ofa particle and the time t at which the particle has that value for the energy. The uncertainty Am in the angular frequency of the wave packet is related to the uncertainty A in the energy of the particle by Am = h.E/h, so that the relation (1.25) when applied to a free particle becomes... 

We have shown in the two examples above that the uncertainty Ax in the position of a wave packet is inversely related to the uncertainty Ak in the wave numbers of the constituent plane waves. This relationship is generally valid and... 

The lower bound applies when the narrowest possible range AA of values for k is used in the construction of the wave packet, so that the quadratic and higher-order terms in equation (1.13) can be neglected. If a broader range of k is allowed, then the product AxAk can be made arbitrarily large, making the right-hand side of equation (1.23) a lower bound. The actual value of the lower bound depends on how the uncertainties are defined. Equation (1.23) is known as the uncertainty relation. 

A similar uncertainty relation applies to the variables t and o . To show this relation, we write the wave packet (1.11) in the form of equation (B.21)... 

This uncertainty relation is also a property of Fourier transforms and is valid for all wave packets. 

Since a free particle is represented by the wave packet I (jc, i), we may regard the uncertainty Ajc in the position of the wave packet as the uncertainty in the position of the particle. Likewise, the uncertainty Ak in the wave number is related to the uncertainty Aj3 in the momentum of the particle by Ak = hsp/h. The uncertainty relation (1.23) for the particle is, then... 

The Heisenberg uncertainty principle is a consequence of the stipulation that a quantum particle is a wave packet. The mathematical construction of a wave packet from plane waves of varying wave numbers dictates the relation (1.44). It is not the situation that while the position and the momentum of the particle are well-defined, they cannot be measured simultaneously to any desired degree of accuracy. The position and momentum are, in fact, not simultaneously precisely defined. The more precisely one is defined, the less precisely is the other, in accordance with equation (1.44). This situation is in contrast to classical-mechanical behavior, where both the position and the momentum can, in principle, be specified simultaneously as precisely as one wishes. 

Fig. 4.2.1 The probability density associated with the Gaussian wave packet. The most probable position is at x = xt, which also coincides with the expectation (average) value of the time-dependent position. The width is related to the time-dependent uncertainty (Ax)t, i.e., the standard deviation of the position.

The relation (1.45) may also be obtained from (1.44) as follows. The uncertainty AE is the spread of the kinetic energies in a wave packet. If Ap is small, then AE is related to Ap by... 

The uncertainty relation can also be deduced from the following general idea. If we propose to build up a wave packet, extending for a finite distance in the cc-direction, from separate wave trains, we... 

The beat pattern consists of what is known as a pilot wave or a wave packet. As indicated in the diagram, one of the ways to localize the wave packet into a narrow region of space is to combine two waves that have vastly different wavenumbers (the wavenumber I /A is directly related to the momentum by a factor of h). Thus, in order to minimize the uncertainty in the position, a wider range of wavelengths (or a greater uncertainty in momentum) is required, as illustrated in Figure 3.22. This is simply a qualitative restatement of the Heisenberg uncertainty principle. 

As a preliminary, we briefly return to the idealized model, where the incident photon states are represented by a pure one-photon quantum state. In this case, the detection signals / " = (S) and defined by Eqs. (73) and (74), are entirely determined by Fourier transformation (72) relating the time function /1 " (S) to the spectral distribution F(E. According to the uncertainty principle in Fourier analysis (Papoulis, 1962), the duration Ate of the intensity signal P " HS) is then related to the spectral width of the wave packet [Eq. (55)] by... 

The SchrOdinger wave thus explains the Heisenberg uncertainty relations. Particle properties are found if the wave packet has a small extension in space. D p then has to be large. On the other hand, it may be shown that the average momentum and average acceleration are that of the classical particle. 

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