Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
The Quantum Mechanics Solver 18 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 | 172 17 A Quantum Thermometer 17.3 Coupling of the Cyclotron and Axial Motions We now study a method for detecting the cyclotron motion. This method uses a small coupling between this motion and the axial motion. The coupling is produced by an inhomogeneous magnetic field and it can be described by the additional term in the Hamiltonian W Mw2 nr z2 . The experimental conditions are chosen such that e 4 x 10 7. 17.3.1. Write the total Hamiltonian Hc H W using the operators hr hi pz and z. 17.3.2. Show that the excitation numbers of the cyclotron motion hr and of the magnetron motion hl are constants of the motion. 17.3.3. Consider the eigensubspace Enr ni of hr and hl corresponding to the eigenvalues hr and hl. a Write the form of Hc in this subspace. b Show that the axial motion is harmonic if the system is prepared in a state belonging to Enr ni. Give its frequency in terms of hr and hl. c Give the eigenvalues and eigenstates of Hc inside Enr ni. 17.3.4. Deduce from the previous question that the eigenstates of Hc can be labeled by 3 quantum numbers hr hl hz. We write these states as hr hl hz . Give the energy eigenvalues in terms of these quantum numbers and of wr Wl p. and e. 17.3.5. One measures the beat between a highly stable oscillator of frequency wz 2n delivering a signal proportional to sin wzt and the current induced in an electric circuit by the axial motion. This latter current is proportional to pz t . a Calculate the time evolution of the expectation values of the position and momentum operators z and pz assuming that the state of the electron is restricted to be in the subspace Enr ni. We choose the initial conditions z t 0 z0 and pz t 0 0. b To first order in e what is the phase difference p between the detected current and the stable oscillator after a time t Show that the measurement of this phase difference provides a measurement of the excitation number of the cyclotron motion. 17.3.6. We now assume that the electron is in an arbitrary state 2 .