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Transcript of Quantum A2
9 Classical Mechanics Euler-Lagrange Hamiltonian Time independent Schroedinger time dependent time indepedent impulse operator hamiltonian operator Time Independent Pertubation Theory Potential well with and without small pertubation Suppose we have solved the (time-independent) Schrodinger equation for some potential and have obtained a complete set of orthonormal eigenfunctions.
Now we perturb the potential slightly. We'd like to find the new eigenfunctions and eigenvalues.
Pertubation theory is a systematic procedure for obtaining approximate soutions to the perturbed problems. We write the new Hamiltonian as the sum of two terms superscript 0 always describes the unperturbed quantity Next we write and as a power series in First order theory First order correction to the Energy is the expectation value of the pertubation in the unpertubed state Note that the denominater is safe as long as the unperturbed energy spectrum is nondegenerate (if it is you divide by zero) Note: pertubation theory yields surprisingly accurate results for the energy but quite poor results for the wavefunctions. Second order energies Non Degenerate Pertubation Theory Degenerate Pertubation Theory Suppose then : Two fold degeneracy Typically a pertubation will break this degeneracy and create two distinct states.
Because we do not know what the good linear combination will be we cannot calculate the first order energy, we do not know what unpertubed states to use. For the moment we write the good unpertubed states in the general form. That is, we keep and adjustable. We want to solve the Schrodinger equation With Plugging in the power series into the schrodinger equation and using the orthonormality condition we find Linear Algebra Hermitian commute Higher-Order Degeneracy Eigen vector Evidently the 's are nothing more than the eigenvalues of the W matrix Finding the good unpertubed wave functions amounts to constructing a basis in the degenerate subspace that diagonalizes the matrix W Example: 3D infinite cubical well a a/2 a a a/2 3D infinite cubical well potential We introduce a pertubation: in the shaded area the potential is raised by The ground state is nondegenerate, the first excited state is triple degenerate First order correction to the ground state is For the first excited state we need to use the degenerate pertubation theory Step 1: Construct the Matrix W the diagonal elements are the same as for the ground state the off-diagonal elements are more interesting but the z integral is zero the most interesting is which gives us the matrix Orthogonal Unitair Eigenvector of T Transformation T Eigenvalue characteristic equation We use the characteristic equation to find the eigenvalues of Step 2: Find the eigenvalues of the matrix, these are the first order corrections to the energy Leading to the following energies Meanwhile the "good" unpertubed states are linear combinations of the form The coefficients are the eigenvectors of the matrix W Thus we find that the "good" states are The Fine Structure of Hydrogen In order to study hydrogen we took the hamiltonian to be kinetic coulombic potential This is ofcourse an approximation; there is a tiny pertubation due to the fine structure The fine structure constant: Bohr energies: of order
Fine structure: of order
Lamb shift: of order
Hyperfine splitting: of order Due to a relativistic correction and spin- orbit coupling Due to the quantization of the electric field Due to the magnetic interaction between the dipole moments of the electron and the proton The Relativistic Correction The first term of the Hamiltonian is supposed to represent the kinetic energy with operator: This is the Classical kinetic energy the relativistic energy is Or in terms of momentum We can expand this about (p/mc) The first order correction to the Hamiltonian is thus In first order pertubation theory the correction is given by the expectation value of the pertubation using the Schrodinger equation Working this out for Hydrogen we find: Spin-Oribit Coupling General Quantum Commutation relations Eherenfest Theorem:
Expectation values obey classical laws Generalised uncertainty principle Recap H-Atom Pertubation Theory Time Dependent Pertubation Theory Quantum Cryptography General Angular Momentum Spin Zeeman Effect Spontaneous Emission Formalism The state of a system corresponds to a vector in Hilbert Space An Observable corresponds to a Hermitian Operator Postulates of QM Time evolution is given by A measurement of will always yield an eigenvalue of For a system in state the probability of eigenvalue with corresponding eigenfunction of is given by After the measurement the wavefunction collapses to the corresponding eigenstate eigenstates of the hydrogen atom
specified by 3 letters using commutation relations we derive eigenvalues and eigenstates, to be contrasted with solving the Schrodinger equation To bad no joint eigenfunctions, so impossible to measure all clearly at the same time. Luckily each component does comute with General Quantum Theory for angular momentum State s = fixed Integer solutions Half integer solutions Pauli Spin Matrices Spin-1/2 particle can be expressed as a two-element column matrix or spinor representing spin up representing spin down Spin operators become 2x2 matrices Matrix Multiplication Larmor Precession Particle with Spin in magnetic field Hamiltonian in Matrix Form with eigenstates Because H is time independent the time dependent solution can be expressed in terms of the stationary states. Because we can write If we know calculate the expectation value of S using the spin matrices we find that S is tilted to the z-axis and precesses at a frequency called the Larmor frequency just as it would classically Addition of Angular Momenta Addition of 2 spin 1/2 particles triplet state singlet state Glebsch Gordon Coefficients Classical (RSA) Encryption Alice Bob Eve Alice Bob Eve Public Channel QM-Channel (private) 2 Encryption 3 Decryption 1 Key generation Public key used for encryption
Private key used for decryption Alice sends the public key to Bob and keeps the private key secret
Bob encrypts message M using the public key and sends it to Alice Alice can recover the message M from the encrypted signal by using her private key Weakness With enough computational power the public key can be broken and the message can be decrypted Quantum Encryption Entanglement An entangled state of 2 particles is a state that cannot be written as the product of 2 one particle states 2 Encryption 3 Decryption 1 Key generation A pair of entangled particles (photons) is send through the private channel
Both Alice and Bob meassure the polarization using a random orientation of their detectors
They communicate the orientations they used through the public channel
When they haven't used the same orientation, the measurement is discarded
The instances where they used the same orientation are used as the key
The Entanglement ensures that the key is the same for Alice and Bob Alice can recover the message M from the encrypted signal by using the key Bob encrypts message M using the key and sends it to Alice If Eve tries to intercept the key Eve is forced to use a certain orientation for her detectors.
The act of detecting colapses the wavefunctions to the basis of the detector
Because it is impossible to always have the same orientation as Alice or Bob Eve can't prevent that the particle she send to Bob is not in the same state as the one she received
When Eve is spying the key Alice and Bob have no longer matches, the message is scrambled and they know they have an eavesdropper Gauranteed secure key Public Key & Encrypted Message Detector orientations & Encrypted Message Entangled Photons Technical Challanges Sustaining the entanglement over longer distances
When working with single photons the signal strength is minimal, there is very little room for disturbances due to glass fibre and noise Example Selection Rules r v proton electron r -v proton electron H-atom Electron point of view Circular current-> magentic field at the location of the electron Magnetic field due to coupling Proton charge mass of electron Angular momentum of electron factor 2 dropped due to g-factor from Quantum electrodynamics factor 2 due to thomas precession as a correction for non inertial refference frame In the pressence of spin-orbit coupling the Hamiltonian no longer commutes with L and S, it does commute with and and the total angular momentum . These are good states to use in pertubation theory. We find: with eigenvalues and the expectation value for Combining gives Atom placed in external magnetic field Weak Field Intermediate Field Strong Field Allowed transitions note! Casimir Effect Spontaneous emission rate Lifetime General Math Series expansion With time dependend pertubation theory we look at the change in occupation of the states. We do not look at the change in energy of the states. we want to know how the occupation of each state changes in time The General solution is with however this is only solvable for a 2 state system Assume that at t = 0 only 1 state is occupied k l Iteration Zeroth Order aproximation Pretend it is 1 Pretend there is no pertubation First Order aproximation Sinusoidal Pertubations Terms are only significant when the denominaters are close to zero. This gives two possibilities (possible when ) (possible when ) System absorbs one energy quant System emits one energy quant Stimulated Emission! Fine structure dominates
Zeeman treated as pertubation Zeeman effect dominates
Fine structure treated as pertubation Full machinery of degenerate pertubation theory needed Quantum Harmonic Ocilator slgh