Lecture 1 The Schrödinger equation In quantum mechanics, the fundamental quantity that describes both the particle-like and wave-like characteristics of particles is wavefunction, Ψ(rr, t). The probability of finding a particle in the infinitesimal volume, dv, about rrin space is dvtrdvtrP2|),(|),(rrΨ= (1.1) where *),(),(|),(|2txtxtx Ψ⋅Ψ=Ψ is measurable and it is just the probability per unit volume or probability density. However, ψ(rr,t) itself is not measurable. The sum of the probability over the entire space must be 1 or .1|),(|2=Ψ∫dvtrspaceentire r (1.2) This equation is called normalization condition. The fundamental problem of quantum mechanics is: Given the wavefunction at an initial time, say t=0, to find the wavefunction at any subsequent time t. This can be carried out by solving the Schrödinger equation, ,),(),(),(222ttritrtrVm ∂Ψ∂=Ψ⎥⎦⎤⎢⎣⎡+∇−rhrrh (1.3) where sJ ⋅×=−341005.1h is the Planck constant, m is the mass of the particle and V(r) is the potential field in which the particle is moving. The Schrödinger equation for collection of particles like many electrons and nuclei in a molecule is very similar. In this case, the wavefunction, ),...,,(321trrrrrrΨ=Ψ, is a function of the coordinates of all the electrons and nuclei in the molecule and time, t. If V in the equation is time independent, we can attempt to separate the time dependent part of ),( trrΨ=Ψfrom the space dependent part by writing)()(),( trtrφψrr=Ψ . (1.4) Substituting 1.4 into 1.3, we obtain two equations, for ψ(r) and φ(t), respectively. The equation for φ(t) is idtdtEthφφ()()= (1.5) and its solution is Ψ(,) ()xt xeiEt=−ψh. The equation for time-dependent wavefunction, ψ(r), is −∇ + =hrrr r222mrVr r Erψψψ() ()() (). (1.6) This is the time independent Schrödinger equation. Since 22|)(||),(| rtrrvψ=Ψ , the probability is time independent. For this reason solutions in separable form are called STATIONARY states - the probability is static and energy is conserved. If denoting )(222rVmHrh)+∇−= (1.7) called Hamiltonian operator or simply Hamiltonian of the particle, then the time-independent Schrödinger equation becomes ).()(ˆrErHrrψψ= (1.8) This is an eigen equation: an equation in which an operator acting on a function produces a multiple of the function itself. The set of functions for which the eigen equation holds are eigenfunctions, describing the different stationary states of the system, and the associated E for the eigenfunction are eigenvalues, corresponding to the energies of the different stationary states. We wish to determine the wavefunction of a molecular system, but we will start with the simple hydrogen atom.The Hydrogen atom The hydrogen atom problem represents one of the few problems in quantum mechanics that can be solved exactly. It is the prototype system for the many complex ions and atoms of the heavier elements. Indeed, our study of hydrogen atom ultimately will enable us to understand the periodic table of the elements. The hydrogen atom problem is also very useful in addressing the problem of doping in semiconductors as well as exciton problem. Finally the concepts such as orbitals and energy levels will be crucial for understanding behavior of molecules. The Hamiltonian of H atom is (1.9) 42ˆ0222remHπε−∇−=h where the first term is the kinetic energy operator and the second term is due to electrostatic attraction between the electron and the nucleus. m in the above expression is the reduced mass of the electron and nucleus, but it is very close to the electron mass since the nuclues (proton) mass is about 2000 times the electron mass. Since the potential energy term depends only on the distance r between the electron and the nuclues, it is easier to solve the Schrodinger eq. in spherical coordinates, which takes the form of ),,(),,()42ˆ12(0222222ϕθψϕθψπεrErremrrrrm=−+∂∂−lh (1.10) where }sin1)(sinsin1{ˆ22222ϕθθθθθ∂∂+∂∂∂∂−= hl (1.11) is associated with the square of the angular momentum. Note that the 2ˆl term contains the angle-dependent of the wavefunction and satisfies),()1(),(ˆ22ϕθϕθmlmlYllY hl += , (1.12) where ),(ϕθmlY is spherical harmonics, l=0, 1, 2, …. , called angular momentum quantum number, m=-l, -l+1, …, l called magnetic quantum number. Accordingly we can write the complete solution of Eq. 1.10 as products of angular and radial eigenfunctions as ),()(),,(ϕθϕθψmlnlnlmYrRr = (1.13) where Rnl is special functions, and n=1, 2, 3, …, is called the principal quantum number, l=0, 1, 2, …. n-1 and m=-l, -l+1, …, l. The energy eigenvalues of H-atom is given are given by eVnaneEn202206.134121−=−=πε (1.14) where ohA53.042200==meaπε is the Bohr radius. Shells and Subshells: All states with the same principal number n are said to form a SHELL. These shells are identified by the letters K, L, M, …, which designate the states for which n=1, 2, 3, … Likewise, states having the same value of both n and l are said to form a SUBSHELL. The letters, s, p, d, f, … are used to designate the states for which l=0, 1, 2, …Hydrogen atom – ground state: The ground state of a one-electron atom with atomic number Z, for which n=1, l=0 and m=0, has energy, eVE 6.131−= and wavefunction, πϕθϕθψ212),()(),,(0/2/300010100areaYrRr−== (1.15) Notice that the wavefunction does not depend on angle, it is spherically symmetric. In fact any state with l=0 is spherically symmetric and called s-state. Hydrogen atom – Lowest Excited states: The lowest excited states are four degenerate states, ψ200, ψ210, ψ211, ψ21-1. All of them have the same principal quantum number, n=2, hence the same energy eVeVE 4.326.1321−=−= and the wavefunctions are 02/02/300020200)2()2(121),()(),,(arearaYrRr−−==πϕθϕθψ : 2s state. (1.16)θπϕθϕθψcos)2(11),()(),,(02/2/500121210arreaYrRr−== : 2p0 state. (1.17) φθπϕθϕθψiarereaYrRr sin)2(181),()(),,(02/2/501121211−−== : 2p1 state. (1.18) φθπϕθϕθψiarereaYrRr−−−−== sin)2(181),()(),,(02/2/501121121 : 2p-1 state. (1.19) The wavefunctions given by 3.10 and 3.11 are often combined to form the so-called 2px and 2py states, xpx~cossin~)(~1212112φθψψψ−− (1.20) ypy~sinsin~)(~1212112φθψψψ−+ . (1.21) Hydrogen atom – Second Lowest Excited states: The second lowest excited states are 9 degenerate states, ψ300,