The Uncertainty Principle

The Heisenberg uncertainty principle, given in 1927, also applies primarily to very small particles, and states that we cannot describe with absolute accuracy the behavior of these subatomic particles. The uncertainty principle describes a fundamental relationship between conjugate variables, including position and momentum and also energy and time.

The first statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle. If the uncertainty in the momentum is ∆p and the uncertainty in the postion is ∆x, then the uncertainty principle is stated as

(1)

where h is defined as h = h/2π = 1.054 × 10^-34 J-s and is called a modified Planck's constant. This statement may be generalized to include angular position and angular momentum.

The second statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy. Again. if the uncertainty in the energy is given by AE and the uncertainty in the time is given by Ar, then the uncertainty principle is stated as

(2)

One way to visualize the uncertainty principle is to consider the simultaneous measurement of position and momentum, and the simultaneous measurement of energy and time. The uncertainty principle implies that these simultaneous measurements are in error to a certain extent. However, the modified Planck's constant h is very small; the uncertainty principle is only significant for subatomic particles. We must keep in mind nevertheless that the uncertainty principle is a fundamental statement and does not deal only with measurements.

One consequence of the uncertainty principle is that we cannot. for example. Determine the exact position of an electron, We will, instead, determine the probability of finding an electron at a particular position. In later chapters, we will develop a probability density function that will allow us to determine the probability that an electron has a particular energy. So in describing electron behavior, we will he dealing with probability functions.

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إعادة النظر في اللامحلية

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الاحتمالات والقياسات

مبرهنة بيل وضرورة اللامحلية

الترابط والتشابك

الحالات الكمية والتراكب الكمي

التداخل والقياس

ألغاز الكم

Equipartition and the quantum oscillator