Ch 28 2 Two Approaches to Quantum Mechanics Schrödinger Wave Equation: is a “wave” equation similar to the equations that describe other waves Heisenberg Method: based on matrices. We will only study one part- the Heisenberg Uncertainty Principle It was soon realized that the two methods gave equivalent results Modern quantum mechanics includes elements of both. Correspondence Principle: required that a new theory must be able to produce the old classical laws when applied to macroscopic phenomena.
Ch 28 3 Heisenberg Uncertainty Principle Consider the problem of trying to “see” an electron with a photon We will refer to the uncertainty in x as x If is the wavelength of the light then from diffraction: Photons have momentum p=h / and when the photon strikes the electron it can give some or all of its momentum to the electron the product of these two is
Ch 28 4 Heisenberg Uncertainty Principle A more careful analysis of this gives There is also an uncertainty principle for energy and time
Ch 28 5 Example The strong nuclear force has a range of about 1.5x m. In 1935 Hideki Yukawa predicted the existence of a particle named the pion (π) that somehow “carried” the strong nuclear force. Assume this particle can be created because the uncertainty principle allows non-conservation of energy by an amount ΔE as long as the pion can move between two nucleons in the nucleus in time Δt so that the uncertainty principle holds. Assume that the pion travels at approximately the speed of light and estimate the mass of the pion. (See page 896 in textbook) If the velocity of the pion is slightly less than c, then it can travel the distance d = 1.5x m in the time Δt where The rest mass energy of the pion is equal to the uncertainty in energy
Ch 28 6 Example 28-1 (continued). The strong nuclear force has a range of about 1.5x m. In 1935 Hideki Yukawa predicted the existence of a particle named the pion (π) that somehow “carried” the strong nuclear force. Assume this particle can be created because the uncertainty principle allows non-conservation of energy by an amount ΔE as long as the pion can move between two nucleons in the nucleus in time Δt so that the uncertainty principle holds. Assume that the pion travels at approximately the speed of light and estimate the mass of the pion. (See page 896 in textbook) A few years later the pion was discovered and it’s actual mass is ≈ 140 MeV/c 2. We substitute the above expressions for ΔE and Δt:
Ch 28 7 Example Estimate the lowest possible kinetic energy of a neutron in a typical nucleus of radius 1.0x m. [Hint: A particle confined to the nucleus will have momentum at least as large as its uncertainty. The size of the nucleus is ] We will assume v ≈Δv. It should be noted that v is about 20% of c, which might require a relativistic calculation, but we just want a rough estimate. Notice that the neutron is in rapid motion in the nucleus.
Ch 28 8 Schrödinger Wave Equation De Broglie had said that electron formed standing waves in the Hydrogen atom Schrödinger found a wave equation that described these waves and provided additional information about the electron states The Schrödinger Wave Equation is a better description than the simple picture of standing circular waves that we saw at the end of last chapter It is a second order differential equation
Ch 28 9 What does solution of “Wave Equation” for the Hydrogen Atom give? Three new “quantum numbers” in addition to n Wave function (Greek letter psi) is the amplitude of the matter wave at any point in space and time 2 is the probability of finding the electron at a given point in space. Same formula for energy levels as Bohr Model
Ch Probability vs. Determinism Newton had introduced a era of determinism—if we knew the position and velocity of an object, we can predict its future behavior Uncertainty Principle says we can’t know this Predictions of wave equations for the position and momentum of electron are based only on probabilities. The kinetic theory of gases is based on probabilities but this is different because gas molecules were assumed to move in a deterministic way but there were too many molecules to keep track of.
Ch Principal Quantum Number n Can have any integer value n = 1, 2, 3,... Determines energy of state with the same formula as the Bohr Model Other quantum numbers have a small effect on energy
Ch Orbital Quantum Number l Related to the angular momentum of electron Must be integer less than n l = 0, 1, 2, 3, …(n-1) Angular Momentum is given by where Example: l = 2
Ch Standard Notation l = 0 is s state l = 1 is p state l = 2 is d state l = 3 is f state (then alphabetically g, h, i, j…) Example: state with n = 3, l = 2 is 3 d state n = 3, l = 1 is 3 p state n = 3, l = 0 is 3 s state
Ch Quantum Numbers Magnetic Quantum Number: m l Integer values from –l to +l Example: if l = 2, then m l = 2, 1, 0, -1, -2 Note that there are 2 l +1 possible values space quantization: the angular momentum can only have integer projections on the z-axis ( L z ) Usually defined by magnetic field.
Ch Evidence for Magnetic Quantum Number m l Integer values from –l to +l Example: Upper State if l=2, then m l = 2, 1, 0, -1, -2 Lower State: l=1, then m l = 1, 0, -1
Ch Spin Quantum Number m s Two possible values m s = + ½ or m s = - ½ Was observed that spectral lines were split, even without magnetic field, so something with two values was needed. Called spin quantum number as if electron was a negatively charged sphere that was spinning We now know this spin picture is not correct We still speak of spin up: m s = + 1/2 spin down: m s = - 1/2
Ch Wave functions Every state has a different wave function 2 nlm is the “probability distribution” Sometimes referred to as electron cloud Probability distribution for ground state of H atom (1s)
Ch Wave functions n=2, l=0, m l = 0 (2s) state n=2, l=1, m l = 0 (2p) state n=2, l=1, m l = ± 1 (2p) state
Ch Other Wave Functions
Ch Photon Selection Rules Quantum Mechanics provides a means of calculating the probability of electron changing state and emitting a photon Photon has spin of 1 so this favors transitions with transitions that obey this are called allowed transitions transitions that do not obey this are forbidden transitions and have a very low probability.
Ch Single Electron Atoms this model explains Hydrogen very precisely, but does it apply to atoms with more electrons and more protons in the nucleus? It works well for all single-electron ions such as He + (Z = 2) and Li ++ (Z = 3). It can be used with adjustments for atoms where there is a single valence electron outside of a closed shell
Ch The Pauli Exclusion Principle Atoms with more than one electron obey Pauli Exclusion Principle: Each electron occupies a particular quantum state with n, l, m l and m s. No two electrons in an atom can occupy the same quantum state Thus no two electrons can have the same set of quantum numbers n, l, m l and m s.
Ch Helium (z = 2) nlmlml msms 100½1s1s 100-½-½1s1s Helium: n=1, l=0 Each electron is attracted to nucleus but repelled by other electron.
Ch Lithium (Z = 3) nlmlml msms 100 ½ ½ 200 ½ Lithium has a single electron in the n =2,l=0 state (2s). This single electron experiences attraction of three protons and the repulsion of the two inner electrons which gives a net charge of approximately e. So the 2s electron is somewhat hydrogen-like and is not tightly bound. We write the electron configuration of lithium as 1s 2 2s n = 1, l = 0 n = 2, l = 0 n = 3, l = 0
Ch Energy Level diagram for He, Li, and Na
Ch Periodic Table of Elements Shell: Electrons with the same value of n. K shell: n =1 L shell: n = 2 M shell: n = 3 Subshell: Electrons with the same n and l values Examples: 2s Subshell 3p Subshell
Ch X-Ray Spectra When we derived the formula for the energy of stationary states, we did not let Z = 1 because the equation with Z works in many (but not all) cases. This formula can not always be used for the electrons in a nucleus with a large Z because it requires a central positive charge and the presence of the other electrons changes the situation. The nuclear charge Z is shielded by other electrons which move rapidly For an innermost electron, the effect of the outer electrons largely cancel out and thus the formula can work. X-Rays are produced when electrons are accelerated through a high voltage and they then knock an electron out of the n = 1 or 2 atomic states
Ch X-Ray Spectra of Molybdenum Consider what happens when an electron is removed from the n = 1 state of the atom molybdenum (Z=42). An electron from the n = 2 or n = 3 drops down to n = 1 and a photon is emitted That electron ‘sees’ a nuclear charge of (Z-1) because the other innermost electron is still there.
Ch Example: X-Ray Spectra of Molybdenum We thus calculate E 21 = 17.2 keV and E 31 = 20.4 keV and and 21 =0.072 nm and 31 =0.061 nm.
Ch Laser Light Amplification by Stimulated Emission of Radiation Laser uses stimulated emission to amplify a light beam Stimulated Emission: an atom in an excited state is stimulated by an incoming photon to drop to a lower state with the emission of a photon. Spontaneous Emission: Process we have already discussed where atom spontaneously drops to a lower state with the emission of a photon.
Ch Stimulated Emission of Radiation Absorption of Photon: photon must have energy equal to E u – E l Stimulated Emission: photon must have energy equal to E u – E l and two coherent photons are emitted. Stimulated photon emitted faster. Coherent: all parts of beam have the same phase Laser uses stimulated emission to amplify light beam
Ch Population Inversion In order to have LASER action, you must have more atoms in the excited state than in the ground state This is called a population inversion because normally the atoms are in the ground state
Ch Population Inversion in a Ruby Laser In a ruby laser, atoms are put in state E 2 by means of a flash bulb. They then decay to E 1 which is a metastable (long-lived) state where they stay until stimulated emission occurs
Ch Ruby Laser Flash bulb creates a population inversion A few photons are emitted spontaneously Many photons escape A few photons emitted along the axis of the ruby cause stimulated emissions which then cause additional stimulated emissions Beam builds up but a few photons pass through partially transparent mirror