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##### PHY206: Atomic Spectra

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**PHY206: Atomic Spectra**Lectures 5 - 8**Lectures 5-8: Outline**• The Schrodinger equation in 3D • Introduction: Wave Equations • Separation of Variables • Radial Schrodinger Equation • The Hydrogen Atom • Energy Levels, Eigenfunctions • Angular Shapes • Quantum Mechanics of the Hydrogen atom • Radiative Transitions • Relativistic Effects Atomic Spectra**Wave Equations**Wave Equation for non-dispersive waves Solution Did you notice something? If you re-write the equation in “operator form” you see that the wave Y is an eigenfuction of an operator, with eigenvalue = 0 ! Wave Equation for non-dispersive waves Atomic Spectra**A hint for relativistic Quantum Mechanics**But the Einstein equation of Special Relativity is: Klein-Gordon Equation: A relativistic QM particle with mass m Atomic Spectra**The non-Relativistic limit:**We just saw that the equation of a freely moving Relativistic QM particle with mass m (without spin!) is: The non-relativistic limit of this equation is reached when we require the NR energy condition: Schrodinger Equation for a freely moving non-Relativistic QM particle Atomic Spectra**Particle in a central potential**The energy of a classical particle moving in a potential V(r) is : In Quantum Mechanics E is an operator acting on the particle wavefunction: If we replace the momentum and position with the corresponding operators: Atomic Spectra**The Schrödinger Equation**The Schrodinger Wave equation: V(x): Potential Energy Complex wave function Continuous functions Probability to find the particle at (x,t) Normalization Let Time independent Schrödinger equation Atomic Spectra**Seperation of Variables in 1D**Step1: time-dependent 1D Schrodinger wave equation: Step2: assume a free particle V(x)=0 and factorization Step3: find the solutions for the 2 differential equations Atomic Spectra**Spherical Coordinates**z q r y f x Polar angle Azimuthal Laplacian: Atomic Spectra**Schrödinger Equation in Spherical Coordinates**3D Cartesian: Spherical coordinate system: Potential energy: Atomic Spectra**Particle in a central potential V(r)**Separate angular and radial parts by multiplying by r2: Angular Part But we know that the operator in the parenthesis is ~L2 : Atomic Spectra**Particle in a central potential V(r)**The Schrodinger equation now has separate Radial and Angular pieces : So we can write: But we know that: Atomic Spectra**Particle in a central potential V(r)**So the Ylm now can be removed: If we absorb the r in the derivative of R and divide by r2 : Obviously the following simple redefinition leads to the Radial Schrodinger equation: Atomic Spectra**Radial Schrodinger Equation:**… leads to the Radial Schrodinger equation: This is basically a motion of a particle with angular momentum L in an effective 1D potential given in the parenthesis. Application: the Hydrogen atom For finite R at r=0 we have u(r=0)=0, also u(r)=0 at infinity Atomic Spectra**H-atom: Energy Levels**Effective Potetial The minimum of the effective potential is at: Bohr Radius Rydberg Energy Atomic Spectra**Physical Consequences**For l=0 the potential minimum tends to –infinity which physically means that our electron in the H-atom should fall in the nucleus. Actually this doesn’t happen in QM due to the Heisenberg Uncertainty principle: The more we localize a particle the higher its kinetic energy (fluctuations). The minimum kinetic energy is given by the uncertainty principle. Atomic Spectra**Significance of the Bohr radius**Now add this extra kinetic energy to the effective potential and find a new approximate minimum for particles with l=0: So the lowest possible state for an electron with l=0 in a Coulomb potential has an average radius equal to the Bohr radius: the electron will not fall in the nucleus! Atomic Spectra**Energy Levels and Radial Eigenfunctions**u(r)0 as rinfinity Polynomial with nr zeros: nr denotes the number of nodes between 0 and inf. u(r)0 as r0 Atomic Spectra**H Atom: quantum numbers**Principal quantum number n=1, 2, 3 … Energy where E0=13.6eV Orbital quantum number l=0, 1, 2, 3 …, n-1 Orbital angular momentum Magnetic quantum number m=-l, -l+1, …, l-1, l Angular momentum in Z-direction • Bound state energies are negative. • Energy dependence only on n is a result of spherical symmetry. • Angular momentum is quantized in magnitude and direction Atomic Spectra**Include spin: J=L+S**As we will see later the spin of the electron in the Hydrogen atom causes a small (about 1/1000) but measurable effect in the energy levels. The total angular momentum J is then given by the sum of the orbital part L and the spin part S. Spectroscopic Notation: Atomic Spectra**H atom: the ground state wavefunctions**Ground state n=1, l=m=0 where Bohr radius Normalization condition Radial probability density Electrons can be anywhere, but most likely to be at r=a0for the ground state. Atomic Spectra**H atom: first excited state wavefunctions**1st excited (n=2) state has three degenerate states n=2, l=0, m=0 n=2, l=1, m=0 n=2, l=1, m=±1 Bohr model: Atomic Spectra**Angular Shape of Wave Functions**“2s state” “2p states” “1s state” n = 1 2 2 l = 0 0 1 ml= 0 0 0, ±1 Atomic Spectra**Parity**Imagine an operator P (called Parity) with the property: The eigenfuctions and eigenvalues of P can be found: So we can only have two eigenvalue equations: Parity + (even function) Parity - (odd function) Atomic Spectra**Parity**For a particle in a central potential: When l = 0,2,4,… (even) then the parity is even (+) When l = 1,3,5,… (odd) then the parity is odd (-) Conservation of Parity: Formal: when the parity operator P commutes with the Hamiltonian Meaning: no observable change in the state of a particle when the coordinates undergo a reflection through the origin Example: Electromagnetic Interactions conserve parity (i.e. they don’t care about reflections!) Atomic Spectra**Radiative Transitions**n=1 n=3 n=4 n=2 S P D F G l=0 1 2 3 4 n=5 Degenerate states Principal quantum number n=1, 2, 3 … with E0=13.6eV Determines energy Atomic Spectra**Interaction with a field**Interaction with a ‘field’ or potential V: No Interaction 1 Interaction 4 Interactions All we need is the probability to go from the initial state, to the final: Atomic Spectra**Electric Dipole Transitions**The hydrogen atom consisting of a proton and an electron is an electric dipole which creates an EM field or a potential: To find transition probabilities between n,l,m and n’,l’,m’ we use the Parity operator 0 for Not 0 for Atomic Spectra**Spontaneous Transitions: the vacuum is not empty**Fields and potentials are generated by (source) particles. For example the presence of a proton in space creates an EM field extending to infinity. What happens if we remove all of these particles and their fields? Vacuum: i.e. empty space However, QM allows vacuum energy to fluctuate according to the Heisenberg Uncertainty Principle: electron positron So, for a sufficiently small time interval Dt we can have enough energy to produce from … “nothing” an electron positron pair. The e-p pair will annihilate to “nothing” after Dt! Atomic Spectra**Spontaneous Transitions**So, the vacuum is boiling by having particles continuously produced and destroyed. These particles produce a real field that can interact with the Hydrogen atom. This interaction can cause transitions from n to n’ with n>n’: Lyman series Balmer series Atomic Spectra**The Reduced Mass Effect**So far we have been assuming MNucleus >> me . The effects of a finite MN are taken into account by considering the reduced mass: Atomic Spectra**Relativistic Effects**We now assume p<<mec and expand the square root about 1: Leading relativistic correction Non-Relativistic Kinetic Energy Rest Mass Atomic Spectra**The fine structure constant**The average momentum of an electron in the hydrogen atom can be estimated by the uncertainty principle Bohr Radius Fine Structure Constant = So, we can use this estimate for the leading relativistic correction: Relativistic correction to the kinetic energy of the electron in the H-atom Atomic Spectra**The spin-orbit interaction**As we have seen electrons have spin S and a magnetic moment given by: But the electron also spins around the proton creating a magnetic field: That gives rise to an interaction of the electron spin to the B-field and an energy: Atomic Spectra**Effect of the SL interaction on the hydrogen atom energy**states The energy due to the spin-orbit interaction for a low lying H-atom state is of order: Where constant f ~ 1/10 Fine Structure Constant = with So, we expect the spin-orbit interaction to be an effect ~1000 times smaller than the ER/n2 levels we extracted from the Schrodinger equation. Atomic Spectra**Examples: n=2, l=0,1 and s=1/2**Spectroscopic Notation: We can experimentally verify the spin orbit effect by measuring an energy difference of: by looking at small difference in the wavelengths of the transitions: Atomic Spectra