## Molecules

### Uncertainty Principle

The Uncertainty Principle is derived from three elements: the wave-particle duality, the indivisibility of energy and momentum transfers, and the lack of complete determinism. It states that for a pair of conjugate variables such as position / momentum and time/energy (including the rest mass energy mc2), it is impossible to have a precisely determined value of each member of the pair at the same time. This statement is illustrated with a schematic diagram in Figure 12-02. The corresponding formula is: x px > , where denotes the uncertainty, x is the position of the point mass m along the x-axis,
px = m vx is the momentum along the x-axis, vx is the velocity along the x-axis, and = h/2 = 1.054x10-27 erg-sec. A similar relation exists for the uncertainty of the time t and energy E, e.g.,

#### Figure 12-02 Uncertainty Principle [view large image] t E  . In case of heavy mass (such as a macroscopic object), the uncertainties and thus the quantum effect becomes very small, classical physics is applicable once more.
See "A Derivation of the Uncertainty Principle" for more detail.
Many quantum phenomena such as superposition, probability density (or wave), vacuum fluctuation, and virtual particles are related to the uncertainty principle:

• By definition, a state consists of all the information needed to completely describe a system at an instant of time. Since the quantum state is specified by momentum, energy, angular momentum, or spin and there is an uncertainty in determining their value, it implies that a particle can occupy many quantum states (with different probability). This is called superposition.
• Figure 12-03a illustrates a very simple superposition of two spin states - one parallel and the other anti-parallel to the direction of a magnetic field, where |a|2 and |b|2 are probabilities of finding the particle in the corresponding state. It is only when the state of the particle, e.g. the spin in this case, is measured that it settles into a definite state. But as soon as we stop monitoring its behavior, the particle dissolves into a superposition again.

#### Figure 12-03a Superposition [view large image]

Note : Probability = |Probability Amplitude|2, e.g., a and b in Figure 12-03a are the probability amplitudes, the corresponding probabilities are |a|2 and |b|2.

• The uncertainty in space and time is interpreted as the probability of finding the particle at a certain time and place. For example, the electron within the confine of the nucleus exhibits a certain probability pattern. Figure 12-03b shows the probability distribution of an electron in the 3d state (ml=0) of the hydrogen atom. When the electron moves in free space with a certain momentum p, the probability pattern displays a wave-like form with the wavelength = h/p, which is known as the de Broglie wavelength. Since there is a spread of momentum according to the uncertainty principle, the electron wave is described by a wave
•  packet, which is the combination from waves of different wavelengths and amplitudes (Figure 12-03c). If an object's wavelength is of a similar order to the size of the objects around it, the wave nature comes to the fore. The wavelength of a macroscopic object such as a moving car is something around 10-36 cm; thus it takes some pretty tiny objects to expose the car's wavelike properties. Only microscopic objects

#### Figure 12-03c de Broglie Wave [view large image]

such as electron has a large enough wavelength to show its wavelike property with objects in manageable size. For example, the de Broglie wavelength for a 75 ev electron is 2 x 10-8 cm, thus the spacing between atoms in a crystal is a good diffraction grating for such electron. • In classical physics, empty space-time is called the vacuum. The classical vacuum is utterly featureless. However, in quantum mechanics, the vacuum is a much more complex entity. It is far from featureless and far from empty. The quantum vacuum is just one particular state of a quantum field. It is the quantum mechanical state in which no field quanta are excited, that is, no real particles are present. Hence, it is the "ground state" of the quantum field, the state of minimum energy. Figure 12-03d illustrates the kind of activities going on in a quantum vacuum. It shows virtual particle pairs appear, lead a brief existence, and then annihilate one another in accordance with the Uncertainty Principle.
• #### Figure 12-03d Vacuum Fluctuation [view large image]

• There are two kinds of virtual particle. One kind are particles produced out of the vacuum as mentioned above. Many such particles can be produced only in virtual pairs (as shown in Figure 12-03d) in order to preserve the existing balance of properties such as electric charge in the Universe. But particles, which are not constrained by these conservation laws, notably photons, can be produced without any mirror image counterpart (such as in the Casimir effect below).

The particles which join the vertices in a Feynman diagram (Figure12-03e) are also virtual particles and can never be detected directly, even though they are of key importance in determining the way "real" particles interact. This kind of virtual particle can be generated in violation of conservation laws, which are obeyed overall during the interaction. All quantum particles can be thought
• of as being surrounded by a cloud of virtual particles (and pairs) of various kinds, which are being created and (usually) reabsorbed by the parent particle. The lifetime of each of these virtual particles (and therefore the distance it can travel from its parent particle) depends on its mass-energy and the leeway allowed by the uncertainty principle. Interactions occur when a real particle come close enough (as in high energy collision) for one or more of the virtual particles in the cloud to be absorbed by the other real particle.

#### Figure 12-03e Feynman Diagram Since the virtual photons in between two parallel metal plates placed a short distance apart can exist only when they can form a standing wave, there are fewer photons in each cubic centimeter of vacuum between the plates than there are in the vacuum outside. So, in effect, there is an excess pressure from outside pushing the plates together. This is known as Casimir effect (see Figure 12-03f). The resulting force is very small, but it has been measured (for plates separated by gaps of a few nanometers), proving that quantum fluctuations of the vacuum are a real phenomenon.

#### Figure 12-03f Casimir Effect [view large image]

• See "Link between Uncertainty and Commutative Relations".

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