Thus for electron exchange the overall wave function for helium in the ground state is. This also means that it cannot take infinite values at any point in space23 (Fig. It has to be continuous (Fig. The wave function is the product of all spinors at sites of the lattice and all metric spinors. Simple energy diagram for molecular orbital wave functions of H2. This also means it cannot take infinite values at any point in space20 (Fig. The postulates of quantum mechanics, especially the probabilistic interpretation of the wave function given by Max Born, limits the class of functions allowed (to “class Q”, or “quantum”). Since the spin state remains antisymmetric, the spatial part must be symmetric with respect to electron interchange. An interchange of electrons makes no change in the spatial wave function. FIGURE 9. In consequence (from the probabilistic interpretation), the wave function is square integrable; i.e., ΨΨ<∞. Notice that for the exchange of electrons, φHe(1, 2) = −φHe(2, 1), which is a condition for Fermi particles. The wave function Ψ is a mathematical expression. From: Modern Aspects of Rare Earths and Their Complexes, 2003, A.A. Ovchinnikov, ... D.V. This practice is followed in many places throughout this book. This is true for shorter internuclear distance R, but does not remain true as the bond is stretched and broken. In general, molecular orbital wave functions lead to a good value for the bond length but a very poor description of the processes of dissociation. 2.5.b). By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. where r is a vector (x, y, z) specifying a point in three-dimensional space. By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an … 19, which we will refer to as the covalent part of the wave function. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. However, Max Born found this interpretation to be inconsistent with the results of scattering experiments. To completely define the wave function, it is necessary to determine the specific form of all site spinors. Dmitriev, in Theoretical and Computational Chemistry, 2002. Show that the square of the modulus of \(\psi(r,t) = \psi(r) e^{-i\,ω-t}\) is time independent. These eigenstates are analogous to the frequency modes of…, …trajectory is replaced by the wave function, a nonlocalized function of space and time. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are square integrable), sometimes with an algebraic structure on the set (in the present case a vector space structure with an inner product), together with a topology on the set. The numerator in Eq. At very large R, the exact wave function will have one electron near the left proton and one near the right, as in Eq. Updates? Klahn and Morgan[56] have examined the convergence in l of model problems while Hill[57] has been able to confirm this rate of convergence for the ground state of the helium atom rigorously. The identification of gTh=ETh∕Δε with the conductance of the system in units of the quantum unit of conductance e2/h was first suggested by Thouless. It is obvious that each index in the formulated wave function is encountered twice, so that the wave function is scalar and, hence, singlet. Each bond has associated with it a metric spinor gλμ with the indices of the ends of this bond. 13, implies that the electrons move independently of each other, i.e., there is no correlation of their motions. Lucjan Piela, in Ideas of Quantum Chemistry (Second Edition), 2014. The probability of rolling a red 3 is 1/6 and the probability of rolling a green 5 is 1/6, so that the total probability of getting both a red 3 and a green 5 is 1/6 × 1/6 = 1/36.

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