Cour 03 Atom in Wave Mechanics PDF

Chapter III: Atomic model in wave mechanics Introduction to the quantum mechanical model of the atom: Thinking about electrons as probabilistic matter waves using the de Broglie wavelength, the Schrödinger equation, and the Heisenberg uncertainty principle III.1 Wave-particle duality...

Chapter III: Atomic model in wave mechanics Introduction to the quantum mechanical model of the atom: Thinking about electrons as probabilistic matter waves using the de Broglie wavelength, the Schrödinger equation, and the Heisenberg uncertainty principle III.1 Wave-particle duality and the de Broglie wavelength Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength λ lambda, given by the following equation: 𝒉 𝒉 𝝀= = 𝒎𝒗 𝑷 Starting from Einstein’s equation, E = m𝑐2 ….. (1) and Planck’s equation E = hν = h/c λ … (2) (1)=(2) h/c λ = m𝑐2 so λ = h /mc Then he generalized the relation for matter by replacing the speed of light c per v (velocity of matter). 𝒉 𝒉 Then the equation becomes: 𝝀 = = 𝒎𝒗 𝑷 It turns out that the wavelike behavior of matter is most significant when a wave encounters an obstacle or slit that is a similar size to its de Broglie wavelength. However, when a particle has a mass on the order of 10−31 kg, as an electron does, the wavelike behavior becomes significant enough to lead to some very interesting phenomena. De Broglie applied his theory of wave--particle duality to the Bohr model to explain why only certain orbits are allowed for the electron. He argued that only certain orbits allow the electron to satisfy both its particle and wave properties at the same time because only certain orbits have a circumference that is an integral multiple of the wavelength of the electron, as shown below. 2πr = nλ 2πr = nh/mv mvr = nh/2π Bohr’s postulate  III.2 Heisenberg’s Uncertainty Principle The Heisenberg uncertainty principle, developed by physicist Werner Heisenberg described the problem by stating: it is inherently impossible to know simultaneously both the exact momentum (mѵ = p) of the electron and its exact location (x) in space. Mathematically, it can be given as in equation: Δ𝑥. Δ𝑝 ≥ ℎ/ 4𝜋 Or: Δ𝑥.Δ(𝑚𝑣)≥ℎ/4𝜋 Or Δ𝑥.Δ𝑣≥ℏ/2𝑚 Δ refers to the uncertainty in that variable ( position x and momentum p ( velocity ѵ) and h is Planck's constant. ℏ : The reduced Planck constant (ℏ=ℎ/2𝜋) From the inequality, we can see that Δx, and Δp are inversely proportional. The inverse proportionality means that as the uncertainty in position decreases, the uncertainty in momentum increases, and vice versa.  Problems 1. The uncertainty in the momentum Δp of a football thrown during the superbowl traveling at 40 m/s is 1×10−6 of its momentum p. What is its uncertainty in position Δx? Mass=0.40kg 2. You notice there is 2 mL of water traveling on the football at the same speed and Δp. Calculate its Δx?. 3. An electron in that molecule of water traveling at the same speed has the same Δp. Calculate its Δx if the mass of an electron is 9.1×10−31kg. 4. Comment on the differences in the uncertainty of momentum between the ball, water, and electron. How does the mass effect this value? Answer 1. 2. The volume is not the property that matters, but the mass. So convert to mass with density.  The mass of 2 ml of water is 2g ( 2.10-3 kg) because the density of water is 1g/ml or 1kg/l 3.  Discussion The mass of the football is 4×10−1kg, the water is 2×10−3kg, and the electron is 9.1×10−31kg. The mass of the water is 2 orders of magnitude smaller than that of the football, and the resulting position uncertainty is 2 orders of magnitude larger. Between the electron and water there is a difference of 28 orders of magnitude for both mass and Δx. There is a direct correlation of inverse proportionality between the Δx and Δp as described by the Heisenberg Uncertainty Principle, and the much smaller electron has a larger position of uncertainty of 1.5 m compared to the larger football's of 3.3×10−30m III.3 Shrödinger’s equation Erwin Schrödinger built upon de Broglie's wave-like concept for particles and formulated wave mechanics, also known as quantum wave mechanics or Schrödinger's wave mechanics. Schrödinger proposed that particles, specifically electrons, could be described by wave functions (mathematical functions that describe the behavior of particles in quantum systems). In 1926, Schrödinger developed a partial differential equation  In classical mechanics, Newton’s second law (F= ma) is used to make a methematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the mementum of the physical system as a fonction of the external force F On the system. Those two parametres are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton’s law is Shrodinguer equation. The concept of a wave function is a fundamental postulate of quantum mechanics, the wave function defines the state of the system at each spatial position and time.  Wave Function (ψ) The wave function, often denoted by Ψ (psi), is a fundamental concept in quantum mechanics. It is a mathematical function that describes the behavior and properties of a quantum system. The wave function, often denoted by Ψ (psi), is a fundamental concept in quantum mechanics. It is a mathematical function that describes the behavior and properties of a quantum system. 𝟐𝝅 𝟐𝝅 y (x,t)=A sin 𝐱. Cos 𝐭 𝝀 𝑻 It can be seen that the amplitude of the wave at any time t is the function of displacement x, the equation for wave motion can be formulate as: 𝑦 = 𝑓(𝑥)𝑓′(𝑡) The symbol of the function 𝑓(𝑥) is replaced by Ψ(x). Then, Ψ(x)=A sin2π/ λ x 𝜕Ψ(𝑥) 2𝜋 2𝜋 Differentiating the above equation = 𝐴 cos x 𝜕𝑥 𝜆 𝜆 𝜕2Ψ(𝑥) 4𝜋2 Differentiating again = − Ψ(x) 𝜕2𝑥 𝜆 𝝏𝟐𝜳(𝒙) 𝟒𝝅𝟐 + Ψ(x)= 0 𝝏𝟐𝒙 𝝀 also, as we know that 𝛌 = 𝐡/ 𝐦𝐯 𝝏𝟐𝜳(𝒙) 4π2m2v2 + Ψ(x)= 0 equation 1 𝝏𝟐𝒙 𝝀  Furthermore, as the total energy E is simply of the potential(V) and kinetic energy E = EK + V  E – V =1/2mv2  mv2 = 2 (E – V) After putting the value of mv2 in equation (1), we get 𝝏𝟐𝚿(𝒙) 𝟖𝝅𝟐𝒎 + (𝑬 − 𝑽)𝚿(𝒙) = 0 𝝏𝒙𝟐 𝒉𝟐 𝒉𝟐 𝝏𝟐𝚿(𝒙) − 𝟐 + VΨ(x) = EΨ(x) equation 2 𝟖𝝅 𝒎 𝝏𝒙𝟐 Equation (2) is Schrödinger equation in one dimension Generalization in three dimensions Extending equation (2) to three dimensions involves considering spatial coordinates x, y, and z. The three-dimensional equation is given by: 𝒉𝟐 𝝏𝟐 𝝏𝟐 𝝏𝟐 − 𝟐 [ 𝟐+ 𝟐 + ] Ψ(x, y, z) + VΨ(x, y, z) = EΨ(x, y, z) … …. (3) 𝟖𝝅 𝒎 𝝏𝒙 𝝏𝒚 𝝏𝒁𝟐 We have: Δ= 𝝏𝟐 + 𝝏𝟐 𝝏𝒙𝟐 𝝏𝒚𝟐 + 𝝏𝟐 𝝏𝒁𝟐 ћ = 2𝜋ℎ  We put the Laplacian operator (Δ) and the reduced Planck constant (ћ) in equation (3). ћ𝟐 We get − ΔΨ(x, y, z) + VΨ(x, y, z) = EΨ(x, y, z) 𝟐𝒎 Hence, the operator for total energy (hamiltonian operator) becomes: ћ𝟐 Ĥ=- Δ + Ep then 𝟐𝒎 ĤΨ(x, y, z) = EΨ(x, y, z) The fondamental equation of the wave mechanics III.4 Solution of the Schrödinger’s equation Solving this equation yields a set of allowed energy states, each associated with a particular eigen value and eigen function. (Ψ1 , E1), (Ψ2, E2), (Ψ3, E3),……………………(Ψn, En) The wave function Ψ is not observable. According to the statistical interpretation of Born, the quantity Ψ∗Ψ = |Ψ2 | is observable and represents the probability density of locating the particle in a given elemental volume.  The solution wave functions of the Schrödinger equation are called atomic orbitals (O.A.) (𝝭2) ; this atomic orbital describes a region of space in which there is a high probability of finding the electron, Any O.A. depends on three quantum numbers noted n, l and ml, explained below for the electron, hence the symbolism 𝝭 nlml and each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that orbital. III.4.1 Principal Quantum Number (n): n = 1, 2, 3, …, 7. Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell (level). For a hydrogen atom with n=1, the electron is in its ground state; if the electron is in the n=2 orbital, it is in an excited state. The total number of orbitals for a given n value is n2 Angular Momentum (Secondary, Azimunthal) Quantum Number (l): l = 0,..., n-1 III.4.2 Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n: III.4.2 Angular Momentum (Secondary, Azimunthal) Quantum Number (l): l = 0,..., n-1 Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n: l 0 1 2 3 4 5... Letter s p d f g h The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f). III.4.3 Magnetic Quantum Number (ml): ml = -l,..., 0,..., +l. Specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on. III.4.4 Spin Quantum Number (ms): ms = +½ or -½. Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).  Number of Electrons Per Sub-shell Per Shell n ℓ m ψn,ℓ,mℓ Subshell Number of 2×(2ℓ+ 1) states n = 1 (K) 0 0 ψ1,0,0 1s 1 2 2 0 0 ψ2,0,0 2s 1 2 8 –1 ψ2,1,–1 n = 2 (L) 1 0 ψ2,1,0 2p 3 6 +1 ψ2,1,1 0 0 Ψ 3,0,0 3s 1 2 –1 Ψ3,1,–1 1 0 Ψ3,1,0 3p 3 6 18 n = 3 (M) +1 Ψ 3,1,1 –2 Ψ3,2,–2 2 –1 Ψ3,2,–1 0 Ψ3,1,0 3d 5 10 +1 Ψ 3,2,1 III.4.4 Representation of Atomic Orbitals s, p and d  s-Orbitals Orbitals are spherical with the nucleus at the center. When n increases, s orbitals become larger and higher in energy because of their increased distance from the nucleus. Shapes of s-orbital  p-Orbitals There are three P orbitals in each P subshell, occur in levels n = 2 and greater. All P orbitals have the same basic shape (two lobes arranged along a straight line with the nucleus between the lobes) but differ in their orientations in space). We denote these orbitals as 2Px, 2Py, and 2Pz. A 2Px orbital has its greatest electron probability along the x-axis, a 2Py orbital along the y-axis, and a 2Pz orbital along the z-axis.  Shapes of p-orbital  d-Orbitals Occur in levels n = 3 and greater. Two fundamental shapes: − Four orbitals with four lobes for each one, centered in the plane indicated in the orbital label dxz, dyz, dxy, and dx2 - y2. − Fifth orbital is uniquely shaped - two lobes along the z-axis and a belt centered in the xy plane dz2. III.4 Diagram of Orbitals each orbital of the subshell ( s, p , d and f) is represented by a box and the electron is represented by an arrow (↑) a positive spin or an arrow (↓) a negative spin. (See the electronic configuration section) s p d f III.6 Energies of Orbitals The energy of an electron in a hydrogen atom is determined solely by the principal quantum n Thus the energy of the orbitals in hydrogen atom increases as follows 1s