A more comprehensive theory was needed. 1925, Schrödinger and Heisenberg separately worked out a new theory Quantum Mechanics.

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1 Ch28 Quantum Mechanics of Atoms Bohr s model was very successful to explain line spectra and the ionization energy for hydrogen. However, it also had many limitations: It was not able to predict the line spectra for other atoms. It could not explain fine structure of emission lines. It could not explain why some spectral line is brighter than others. A more comprehensive theory was needed. 1925, Schrödinger and Heisenberg separately worked out a new theory Quantum Mechanics. Quantum Mechanics is the basic theory at the atomic level. It has been extremely successful to explain a wide range of natural phenomena, and from its predictions, many new practical devices have become possible The Wave Function QM is a statistical theory rather than a deterministic one. In QM, wavelength of particles such as electron is given by de Broglie s formula, λ = h/mv. The amplitude of a particle wave is called the wave function, Ψ(x,t). Ψ(x,t) is calculated by solving the Schrödinger wave equation (not covered here). The meaning of a wave function: Ψ 2 (x,t) represents the probability of finding the particle at the given position x and time t.

2 The classical Newtonian view of the world: if the position and velocity of an object are known at a particular time, its future position can be precisely predicted if the forces on it are known. The quantum mechanics view of the world: the future motion of an object can not be predicted precisely, but the probabilities that an object will be observed at various locations at a given time, can be calculated The Heisenberg Uncertainty Principle Any measurement involves some uncertainty or error. We expect that by using more precise instruments, the uncertainty can be made indefinitely small. However, according to QM, there is actually a limit to the accuracy of certain measurements. This limit does not depend on how well instruments can be made. It results from (1) the unavoidable interaction between the observed object and the observer, and (2) the wave-particle duality. It is not possible to make a measurement of an object without somehow disturbing it. Imagine measuring the position of an electron with light (photon). The resolution limit is about the wavelength of the photon. (a) Higher resolution short wavelength of photon higher energy of photon (E = hf) larger momentum disturbance.

3 (b) Lower resolution longer wavelength of photon lower energy of photon (E = hf) less momentum disturbance. x λ, p h/λ, x p h Heisenberg Uncertainty Principle: Δx Δp Both the position x and momentum p of an object cannot be measured precisely at the same time. The product of the uncertainties ( x p), cannot be less than. = h 2π = J s = J s 2π More accurately the position is measured (i.e., small x), the greater is the uncertainty in momentum, p. More precisely the momentum is determined (i.e., small p), the greater is the uncertainty in position, x. Another from of the uncertainty principle: ΔE Δt, Energy can be uncertain or non-conserved, by an amount E for a time t / E). Example: An electron moves with a constant velocity v = m/s which has been measured to a precision of 0.1%. What is the maximum precision with which its position could be simultaneously measured? The momentum of the electron: p = mv = ( kg)( m / s) = kg m / s

4 The uncertainty is 0.1%, thus Δp = kg m / s From the uncertainty principle: Δx = Δp = J s kg m / s = m Example: J/ψ mesons has very short lifetime before it decays into other particles. The uncertainty of its mass is 63 kev/c 2. Estimate its lifetime using the uncertainty principle. The uncertainty in mass is an uncertainty in its rest energy: ΔE = (Δm)c 2 = ( ev)( J / ev) = J We expect its lifetime to be τ Δt = ΔE = J s J = s 28-5 Quantum Mechanical View of Atoms Electrons do not have well-defined circular orbits as in the Bohr theory, but instead exist as a "cloud". The electron clouds can be interpreted as an electron wave spread out in space, or as a probability distribution for electrons considered as particles.

5 28-6 Quantum Mechanics of the Hydrogen Atom According to quantum mechanics, the state of an electron in an atom is specified by four quantum numbers: n, l, m l, and m s. The Principal Quantum Number, n: Energy levels: E n = 13.6eV n 2, n = 1, 2, 3, n is similar to the quantum number in old Bohr theory. The Orbital Quantum number, l: l can take on integer values from 0 to (n-1). For example, if n = 2, l can be 0 or 1. The magnitude of the angular momentum of the electron L is L = l(l +1) The value of l has almost no effect on the total energy in hydrogen atom. But for other atoms, the total energy depends on both n and l. The Magnetic Quantum Number, m l : m l describes the direction of the electron s angular momentum. It can take on integer values ranging from l to +l. For example, if l =2, m l can be -2, -1, 0, 1, 2. The direction of the angular momentum is specified by giving its component along the z axis (arbitrary). L z = m l,

6 Zeeman effect: When a gas-discharge tube is placed in a magnetic field the spectral lines were split into several very closely spaced lines. Angular momentum of an electron is related to its magnetic moment. Magnetic moment interacts with external magnetic field. Thus the energy level of an electron also depends on m l when a magnetic field is applied. The Spin Quantum Number, m s : An electron can have two possible spin states: spin up (m s = ) and spin down (m s = ). Fine Structure: A careful study of the spectral lines of hydrogen showed that each actually consisted of two (or more) very closely spaced lines even in the absence of an external magnetic field. Fine structure results from spins of electrons interact with the magnetic field due to the orbiting charge. Quantum Numbers for an Electron: Principal n 1, 2, 3,, Orbital l For a given n, l can be 0, 1, 2,, n-1 Magnetic m l For given n and l, m l can be l, l-1,, 0,,-l Spin m s For a set of n,l,m l : m s can be or - 1 2

7 2 possible QM states for n = 1. n = 1, l = 0, m l = 0, m s = 1/2 or 1/2 8 possible QM states for n = 2. n l m l m s possible QM states for n = 3. n l m l m s n l m l m s Generally, the total number of possible states is 2n 2.

8 Example: Calculate E and L for n = 3. Energy: E 3 = 13.6eV n 2 = 13.6eV 3 2 = 1.51eV Angular momentum: L = l(l +1) For l = 0, L = 0 For l = 1, L = 1(1+1) = 2 For l = 2, L = 2(2 +1) = 6 The Selection Rule: When a photon is emitted or absorbed, transitions can occur only between states with values of l that differ by one unit: l = ±1 For example: l = 2 l = 1 or l = 2 l = 3 are allowed. But l = 2 l = 0 or l = 2 l = 2 are not allowed (very low probability).

9 28-7 Complex Atoms; the Exclusion Principle Atomic number Z: the number of proton in the nucleus, which is also the number of electrons in a neutral atom. The Pauli Exclusion Principle: No two electrons in an atom can occupy the same quantum state. Ground state: the lowest possible energy state. Helium (Z = 2): Lithium (Z = 3): n l m l m s n l m l m s Sodium (Z = 11): n l m l m s n l m l m s

10 28-8 The Periodic Table of Elements Shell: Electrons with the same value of n are said to be in the same shell. All electrons with n = 1 are in the first (K )shell. All electrons with n = 2 are in the second (L) shell. All electrons with n = 3 are in the third (M) shell. Subshell: Electrons with the same values of n and l are said to be in the same subshell. According to the Pauli exclusion principle, there can be at most 2(2l+1) electrons in any l subshell. Value of l Symbol Max. No. of electrons 0 s 2 1 p 6 2 d 10 3 f 14 4 g 18 5 h 22 Electron Configuration of some elements: H (Z = 1) 1s 1 He (Z = 2) 1s 2 Li (Z = 3) 1s 2 2s 1 Be (Z = 4) 1s 2 2s 2 Be (Z = 5) 1s 2 2s 2 2p 1 C (Z = 6) 1s 2 2s 2 2p 2 N (Z = 7) 1s 2 2s 2 2p 3. Na (Z = 11) 1s 2 2s 2 2p 6 3s 1