The Quantum Mechanics Solver 23 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 224 21 A Quantum Box -3 -2 -1 A E ha -- 4 o 3 --2 1 Lz h 0 12 3 Fig. . Allowed quantum numbers for the couple Lz E correspond to the degeneracy of an energy level of H0 found previously. This justifies the fact that hi nr form a CSCO. If two different states would correspond to the same couple of eigenvalues n nr the corresponding point of the diagram would be twofold degenerate and the degeneracy of the energy level En would be larger than N 1. . We must find in this subspace two eigenvectors of Lz corresponding to the two eigenvalues h. A first method for finding these eigenvectors consists in calculating the action of Lz on the vectors of the basis nx ny . Using the expression of Lz in terms of ax ay . one finds Lz nx 1 ny 0 ih nx 0 ny 1 Lz nx 0 ny 1 -ih n 1 ny 0 0 -ih or the 2 x 2 matrix to diagonalize . ih 0 j The eigenstates associated to the eigenvalues h are therefore nx 1 ny 0 i nx 0 ny 1 V2 . Another method consists in starting from the ground state nl 0 nr 0 and letting act on this state i the operator aj in order to obtain the eigenvector of energy 2hu and angular momentum h ii the operator a in order to obtain the eigenvector of energy 2hw and angular momentum h. Of course we recover the previous result. Solutions 225 Section Quantum Box in a Magnetic Field . By expanding HB one finds a pI P2j p q2B2 V 2 Lz Hb i . fix . If we set i 2 2 4 we can rewrite HB H cLz 2 where H n is the Hamiltonian of a two-dimensional oscillator of frequency i H px py .2 a2 Ho 2p 2 X y One can then repeat the method of the previous section by replacing by i in the definition of the operators ax ay . One constructs an eigenbasis common to H n and Lz which we continue to note nl nr the eigenvalues being hi ni nr 1 and mh. Each vector nl nr is also an eigenvector of HB corresponding to the energy Em nr hQ n nr 1 hwc nr - ni 2 nr Q ni hQ . . a Two limiting regimes of the magnetic field can be considered corresponding to the limits c C very .
