Synopsis • Review Quantum Mechanics Basic ideas and concept of quantization. • Separation of variables Use separation of variables for solving the Schr¨odingerequation with … Synopsis • Review Quantum Mechanics Basic ideas and concept of quantization. • Separation of variables Use separation of variables for solving the Schr¨odingerequation with cartesian coordinates. Periodic boundary conditions, confined electrons in potent ial wells, simple harmonic oscilla- tor. • Landau levels Charged particles in magnetic field. The Quantum Hall Effect. • Variational Method Principles and applications to simple physical systems. • Perturbation Theory Lowest order time independent and time dependentperturbat ion theory. Apply the results to simple physical situations. • Matrix Mechanics Description of electron spin in terms of Paulispin matrices and associated problems

2.5. General solution of the Schr¨ odingerequation[5] 15 for l =0,1,2,…. Similarly, one can solve for the other two directions and the total eigenvalue of the energy is given by E lmn = ( l +1/2) ~? 1 + ( m +1/2) ~? 2 + ( n +1/2) ~? 3 for l , m , n =0,1,2,3,…. You can consider the SHO in one or two dimensions or combine it with a box or periodic boundary conditions. 2.5 Generalsolution of the Schr¨ odingerequation[5] The time independent Schr¨odingerequation is given, in terms of the Hamiltonian, by H * n ( r ) = E n * n ( r ) where* n and E n , for integer n , are the eigenstates and eigenvalues of the energy. A natural questionarises: if we know the eigen vectors of the Hamiltonain what is the general solution of the Schr¨odinger equation? For that consider its time dependent form H * ( r , t ) = i ~ ? ? t * ( r , t ) This is a differential equation linear in*. From linearity one deduces that the general solution can be written in the form * ( r , t ) = X n a n * n ( r ) e ? iE n t /~ where a n are arbitrary complex numbers. Imposing normalization oft he wave function* ( r , t ) we have the condition P n | a n |2=1. By substituting in the time dependent Schr¨odingerequation one finds that X n a n H * n ( r ) e ? iE n t /~ = X n a n i ~ ? ? t h * n ( r ) e ? iE n t /~ i But we know that H * n ( r ) = E n * n ( r ) and i ~ ?* n ( r ) ? t = E n * n ( r ). Thus, the equation holds identically for arbitrary a n . Quiz Can we interpret: (a) * n as a basis vector? (b) * as a general vector? (c) Space of eigenstates as a vector space? (Hilbertspace) (d) R *?* dV (=1,0) as a”scalar product”?

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