Ch 125a Problem Set 1

Transcription

1 Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract quantum state vector, and to understand it in real space, we can project it as r r φ = φ (r ) r (normally, we drop the basis r, andjustkeepthescalarpartφ (r )) For the inner product φ φ, toevaluateitinrealspace,weinserttheidentityoperator ˆ, which can be expressed in terms of real space projection ˆ = r r dr (the completeness relation), φ φ = φ ˆ ˆ ˆ φ = φ r r φ dr = φ (r ) φ (r ) dr, so we have the integral form of the inner product Note the last step uses the fact that φ r is the Hermitian conjugate of r φ ψ and φ are orthogonal if ψ φ =0 Writethisoutintheintegral form Consider a projection operator ˆP = ψψ, andwrite ˆP φ out in the integral form 3 Any Hamiltonian can be expressed in terms of projection operators of its orthonormal eigenvectors For example, the Hamiltonian of hydrogen atom can be expanded as H = n,l,m E n nlmnlm Using the above projector form, show that H 00 = E 00, 00 H 00 = E and 00 H 00 =0

2 Problem : A wavefunction of optimal size Consider the unnormalized three-dimensional trial wavefunction for the Hydrogen atom defined by r ψ(r) = r< 0 r Hint: we evaluate ψ ψ as follows: ψ ψ = ˆ r0 0 4πr dr r = π 5 r3 0 Calculate the expectation value for the kinetic energy of this wavefunction Verify that it varies as Calculatethe expectation value for the potential energy of the wavefunction ˆV = Verifythatitvariesas r 3 Find the value of that minimizes the total energy Is the resulting energy higher or lower than the exact energy of a hydrogen atom? Why would you expect this? 4 On the same set of axes, plot ˆT, ˆV,andE with respect to What value of minimizes the kinetic energy? What wavefunction shape does this correspond to? What value of minimizes the potential energy? What wavefunction shape does this correspond to? Problem 3: Angular momentum commutators The angular momentum operators are, in position space, L x = yp z zp y, L y = zp x xp z, L z = xp y yp x

3 Use the canonical commutation relation of position and momentum to prove [L x,l y ] = il z, [L y,l z ] = il x, [L z,l x ] = il y Use the above relation to prove L,L z =0 So H, L,andL z form a mutually commutative operator set, and letting us use n, l and m z to uniquely label atomic orbitals Problem 4: Two-state system The Hamiltonian for a two-state system is given by H = a ( + + ), where a is a number with the dimension of energy Assume states and are orthonormal What are,,, and? h h Write down a matrix representation for H (ie H = h h Hint: calculate H, H, H, and H 3 The Schrödinger equation for this Hamiltonian can now be written as the eigenvalue equation h h c c = E h h c c ) Rewrite E as a constant times the identity matrix, then subtract the right hand side from both sides of the equations to get an equation that looks like: c =0 Find the characteristic equation of this matrix via its determinant Solve for the matrix eigenvalues by finding the roots of this equation 4 Plug in each eigenvalue and solve the systems of equations for the respective eigenvectors (as linear combinations of and ) 3 c

4 Problem 5: Raising and lowering operators: quantum harmonic oscillator Given a eigenstate ψ of a Hamiltonian H, raisingandloweringoperators provide a way to get all the other eigenstates For example, consider the quantum harmonic oscillator with Hamiltonian H = ˆp + ωˆx Let s define the raising and lowering operators ω â + = ˆx iω ˆp ω â = ˆx + iω ˆp Show H = ω â + â + Show [â, â + ]= 3 Use the results of and above to prove that if we have Hψ = Eψ, then H(â + ψ)=(ω + E)(â + ψ) This means that given an eigenstate ψ, wegetcanproceedupanddown the energy spectrum to get all the other eigenstates, just by applying the operators â + and â Problem 6: One electron, many nuclei Consider linear H + 3 : 4

5 What is the full Hamiltonian for this system (before the Born-Oppenheimer approximation has been made)? What is the electronic Hamiltonian for the system? Explain why we may normally neglect nuclear motions on the timescale of electronic motions Considering a basis of only s orbitals labeled A, B, andc (centered on H A, H B,andH C ), what are the LCAO wavefunctions appropriate for finite R AB = R BC?Orderthesestatesbyenergyandcallthemφ 0 (ground state), φ (first excited state), and φ (second excited state) The wavefunctions need not be normalized, but remember to satisfy symmetry and orthogonality, using the nodal theorem as a construction guide Label each state with g or u to indicate its symmetry with respect to inversion You can write the final coefficients in terms of S AB = A B, S BC = B C,andS AC = A C 5