Kinetic energy Heisenberg uncertainty principle

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. 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

Extraordinarily small values of G(r )/p(rJ, of the order of 0.03 au, are exhibited by the non-nuclear maxima, the pscudoatoms, in the metallic clusters of Li and Na atoms illustrated in Fig. 2.11. The same small values of kinetic energy per electron are reflected in the ratio of the average values of r(f2) to Al(f2) for the pseudoatoms and, in accordance with the Heisenberg uncertainty principle, they indicate that the charge density of the pseudoatoms is loosely bound and unconfined. The Laplacian distributions for these... 

In reality, it is impossible to achieve a temperature of absolute zero, but this temperature may be approached. We also cannot strictly say that all movement would stop because the Heisenberg Uncertainty Principle (See Skill 1.2a) states that the exact position and momentum of particles cannot be found at the same time. At absolute zero, molecules have the least amount of kinetic energy and motion permitted by the laws of physics. However, this level of being strict is mostly important to physicists and often isn t mentioned in high school chemistry where the kinetic molecular theory is applied. 

The Heisenberg uncertainty principle explains several interesting features of atoms. For instance, electrons cannot exist in planar orbits around the nucleus, as is so commonly depicted by the Bohr model of the atom. The reason for this is because in a planar orbit the uncertainty in position perpendicular to the plane is zero and therefore the momentum in that direction would become infinite. Likewise, the uncertainty principle can explain why the electron in a hydrogen atom does not collapse into the nucleus despite the fact that there is a strong electrostatic attraction in that direction. As the electron s orbit gets smaller, so does the uncertainty in its position. Therefore, the uncertainty in its momentum (and also in its kinetic energy) must necessarily increase. 

Applying the Heisenberg uncertainty principle to the hydrogen atom, we see that in reality the electron does not orbit the nucleus in a well-defined path, as Bohr thought. If it did, we could determine precisely both the position of the electron (from its location on a particular orbit) and its momentum (from its kinetic energy) at the same time, a violation of the uncertainty principle. 

The Heisenberg uncertainty principle is captured by the fact there are many different choices for/(0,x). Equivalently, the quantum-classical correspondence is not unique this is why, for example, there are two (and even more) equivalent forms for the kinetic energy (Equations 1.47 and 1.48). All/(0,x) that are consistent with Equation 1.54 provide a suitable classical correspondence the most popular choice,y(0,T) = 1, corresponds to the Wigner quasi-probability distribution function. ... 

It is possible to show that, for L oo, our particle in a box would have to violate the Heisenberg uncertainty principle to achieve an energy of zero. For, suppose the energy is precisely zero. Then the momentum must be precisely zero too. (In this system, all energy of the particle is kinetic since F = 0 in the box.) If the momentum px is precisely zero, however, our uncertainty in the value of the momentum Ispx is also zero. If Ispx is zero, the uncertainty principle [Eq. (2-46)] requires that the uncertainty in position Ax be infinite. But we know that the particle is between x = 0 and x = L. Hence, our uncertainty is on the order of L, not infinity, and the uncertainty principle is not satisfied. However, when L = oo (the particle is unconstrained), it is possible for the uncertainty principle to be satisfied simultaneously with having E = 0, and this is in satisfying accord with the fact that E = goes to zero as L approaches infinity. 

An electron s kinetic energy in a hydrogen atom is of the order 10 eV. Using the Heisenberg uncertainty principle, determine the linear size of the hydrogen atom. 

How do kinetic isotope effects come about Even in its lowest energy state a covalent bond never stops vibrating. If it did it would violate a fundamental physical principle, Heisenberg s uncertainty principle, which states that position and momentum cannot be known exactly at the same time a nonvibrating pair of atoms have precisely zero momentum and precisely fixed locations. The minimum vibrational energy a bond can have is called the zero point energy (Eo) - given by the expression Eq = j/iv. 

This can be explained by the Uncertainty Principle (Heisenberg principle), which says that if Ax is the uncertainty in a particle spatial location (i.e., also the electron), it is inversely proportional to the uncertainty in determining the particle momentum (or velocity) of AP. Thus, Ax is quite large due to the greatly diffused wave associated with the valence electrons of the metallic bond and results in a decreased AP (from the uncertainty principle), i.e., in the kinetic energy (energy of motion) of valence electrons. 

---