The Heisenberg Uncertainty Principle

The uncertainty principle rationalizes our inability to observe the momentum and position of an atomic particle simultaneously. The act of observation interferes with atomic particles so that their momenta and positions are altered. 

An interesting point to note in the Particle-in-a-Box model problem is that the ground-state energy is not zero as would be predicted by classical mechanics. The physical reason for this paradox has to do with uncertainties in knowing both the position and the momentum of the particle simultaneously due to the wavelike properties of the particle. 

There is inherent error in any type of measurement. The standard deviation is an average range of measurements in a series of trials. As an example, suppose the following values were obtained for some measurement 6.3, 6.8, 6.5, 6.2, and 6.9. The average value is 6.5. The individual trials deviate from this average by -0.2, 0.3, 0.0, -0.3, and 0.4. Simply taking an average of these deviations will result in some 

The analogous approach can be done in quantum mechanical systems. The square of the difference of the operator for an observable, O, from the average, o , is taken O- o y. The uncertainty squared, (AO), is the expectation value of this operator. 

This expression can be simplified by expansion. The o corresponds to a constant which can factored out of the integration. Assume the wavefunction is normalized. Then 

Problem Determine the uncertainty in the momentum, Ap, for the ground-state energy of the Particle-in-a-Box model problem. 

Consider two mechanical quantities A and B, for which the corresponding Her-mitian operators (constructed according to Postulate II), A and B, give the commutator [A, B] = AB— BA = iC, where C is a Hermitian operator. This is what happens for example for A = x and B = px- Indeed, for any differentiable function l one has [ir, px] j = —xih f + ih(x4 ) = ih f , and therefore the operator C in this case means simply multiplication by h. 

In 1937 Werner Heisenberg was at the height of his powers. He was nominated professor and got married. However, just after returning from his honeymoon, the rector of the university called him, saying that there was a problem. In the SS weekly an article by Prof. Johannes Stark (a Nobel Prize winner and faithful Nazi) was about to appear claiming that Professor Heisenberg is not such a good patriot as he pretends, because he socialized in the past with Jewish physicists... 

Wemer Heisenberg did not cany out any formal prool instead he analyzed a Gedankenexperiment (an imaginaiy ideal experiment) with an electron interacting with an electromagnetic wave ( Heisenberg s microscope ). 

Recall the definition of the variance, or the square of the standard deviation of measurements of the quantity A  

FIGURE 5.11 If it is possible to detect which of two interfering paths an electron actually takes, the interference vanishes—no matter how carefully the apparatus is designed to minimize perturbations. 

FIGURE 5.12 Two waves at different frequencies will constructively interfere and destructively interfere at different times. 

FIGURE 5.13 The sum of a large number of sine waves, with a distribution of frequencies of 5% around the center frequency, produces constructive interference for a range At 5 cycles 

The waves produce a pulse of radiation with a finite width (represented by the arrow in the figure). We will define the uncertainty At as the half width at half maximum thus we would use half the length of this arrow, or approximately five cycles at the center frequency vo. The pulse is large as long as most of the frequency components constructively interfere, then grows small. From the figure, At is approximately five cycles at the center frequency vo. Each cycle lasts for a time 1/vo. The net result is  

Equation 5.38 turns out to be a universal result. If we increase the range of frequencies, then the waves constructively interfere for a shorter time. In order to have a single, well-defined frequency (Av 0) the wave needs to continue for a very long time (At - oo). 

The Heisenberg Uncertainly Principle arises from the dual nature (wave-particle) of matter. It states that there exists an inherent uncertainty in the product of the position of a particle and its momentum, and that this uncertainty is on the order of Planck s constant. 

Here s the story with the Heisenberg uncertainty principle the more we know about the momentum of any particle, the Less we can know about the position. The amount of uncertainty is very small on tire order of Planck s constant (6.63 X 10-L+ J s). There an oilier quail lilies besides position and momentum to which the uncertainty principle applies, but position and momentum is the pair that you are likely to need to know for the MCAT. 

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The Heisenberg uncertainty principle offers a rigorous treatment of the qualitative picture sketched above. If several measurements of andfi are made for a system in a particular quantum state, then quantitative uncertainties are provided by standard deviations in tlie corresponding measurements. Denoting these as and a, respectively, it can be shown that... 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

From the Heisenberg uncertainty principle as stated in Equation (1.16) estimate, in cm and Hz, the wavenumber and frequency spread of pulsed radiation with a pulse length of 30 fs, typical of a very short pulse from a visible laser, and of 6 ps, typical of pulsed radiofrequency radiation used in a pulsed Fourier transform NMR experiment. 

Early in the twentieth century physicists established that molecules are composed of positively charged nuclei and negatively charged electrons. Given their tiny size and nonclassical behavior, exemplified by the Heisenberg uncertainty principle, it is remarkable (at least to me) that Eq. (1) can be considered exact as a description of the electrostatic forces acting between the atomic nuclei and electrons making up molecules and molecular systems. Eor those readers who are skeptical, and perhaps you should be skeptical of such a claim, I recommend the very readable introduction to Jackson s electrodynamics book [1]. 

In the earlier treatment we reached the conclusion that resonance absorption occurs at the Larmor precessional frequency, a conclusion implying that the absorption line has infinitesimal width. Actually NMR absorption bands have finite widths for several reasons, one of which is spin-lattice relaxation. According to the Heisenberg uncertainty principle, which can be stated... 

The interpretation of the square of the wave function as a probability distribution, the Heisenberg uncertainty principle and the possibility of tunnelling. 

The difficulty will not go away. Wave-particle duality denies the possibility of specifying the location if the linear momentum is known, and so we cannot specify the trajectory of particles. If we know that a particle is here at one instant, we can say nothing about where it will be an instant later The impossibility of knowing the precise position if the linear momentum is known precisely is an aspect of the complementarity of location and momentum—if one property is known the other cannot be known simultaneously. The Heisenberg uncertainty principle, which was formulated by the German scientist Werner Heisenberg in 1927, expresses this complementarity quantitatively. It states that, if the location of a particle is known to within an uncertainty Ax, then the linear momentum, p, parallel to the x-axis can be known simultaneously only to within an uncertainty Ap, where... 

The location and momentum of a particle are complementary that is, both the location and the momentum cannot be known simultaneously with arbitrary precision. The quantitative relation between the precision of each measurement is described by the Heisenberg uncertainty principle. 

This relationship is known as the Heisenberg uncertainty principle. 

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. 

Using expectation values, we can derive the Heisenberg uncertainty principle introduced in Section 1.5. If we define the uncertainties Ax and Ap as the standard deviations of x and p, as used in statistics, then we have... 

The integrated part vanishes because goes to zero faster than 1 js/ x, as x approaches ( ) infinity and the remaining integral is unity by equation (2.9). Taking the square root, we obtain an explicit form of the Heisenberg uncertainty principle... 

This general expression relates the uncertainties in the simultaneous measurements of A and B to the commutator of the corresponding operators A and B and is a general statement of the Heisenberg uncertainty principle. 

Werner Heisenberg (1901-1976 Nobel Prize for physics 1932) developed quantum mechanics, which allowed an accurate description of the atom. Together with his teacher and friend Niels Bohr, he elaborated the consequences in the "Copenhagen Interpretation" — a new world view. He found that the classical laws of physics are not valid at the atomic level. Coincidence and probability replaced cause and effect. According to the Heisenberg Uncertainty Principle, the location and momentum of atomic particles cannot be determined simultaneously. If the value of one is measured, the other is necessarily changed. 

In the 1920s it was found that electrons do not behave like macroscopic objects that are governed by Newton s laws of motion rather, they obey the laws of quantum mechanics. The application of these laws to atoms and molecules gave rise to orbital-based models of chemical bonding. In Chapter 3 we discuss some of the basic ideas of quantum mechanics, particularly the Pauli principle, the Heisenberg uncertainty principle, and the concept of electronic charge distribution, and we give a brief review of orbital-based models and modem ab initio calculations based on them. 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

One of the first uses of ESR spectra to measure the rate of a chemical reaction was by Ward and Weissman in the early 1950s.1 They made use of a form of the Heisenberg uncertainty principle (eqn 5.1) to relate the lifetime of a spin state to the uncertainty in the energy of the state. 

The Heisenberg uncertainty principle dictates that energy and time associated with an atomic-scale system cannot be determined jointly within arbitrary precision ... 

For matter waves hk is the particle momentum and the uncertainty relation AxAp > h/2, known as the Heisenberg uncertainty principle. 

The Bohr model is a determinant model of an atom. It implies that the position of the electron is exactly known at any time in the future, once that position is known at the present. The distance of the electron from the nucleus also is exactly known, as is its energy. And finally, the velocity of the electron in its orbit is exactly known. All of these exactly known quantities—position, distance from nucleus, energy, and velocity—can t, according to the Heisenberg uncertainty principle, be known with great precision simultaneously. 

The Heisenberg Uncertainty Principle, describing a dispersion in location and momentum of material particles that depends inversely on their mass, gives rise to vibrational zero-point energy differences between molecules that differ only isotopically. These zero-point energy differences are the main origin of equilibrium chemical isotope effects, i.e., non-unit isotopic ratios of equilibrium constants such as K /Kj) for a reaction of molecules bearing a protium (H) atom or a deuterium (D) atom. 

The residual energy (designated of a harmonic oscillator in the ground state. The Heisenberg Uncertainty Principle does not permit any state of completely defined position and momentum. A one-dimensional harmonic oscillator has energy levels corresponding to ... 

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