Lecture 29 Quantum Mechanics (Continued)：29讲量子力学（续）.doc

Lecture 29: Quantum Mechanics (Continued)Reviewo Tunnelingo STMo Imaging WavefunctionsTodayo Harmonic Oscillatoro Hydrogen Atomo Wavefunctionso New Quantum Numbers (l,m)o Eigen-values (energies)Harmonic OscillatorConsider simple harmonic oscillator again. Here we consider two masses m1 and m2 attached to an elastic spring. The potential energy of the system depends on the degree of stretching or compression:We can then immediately write down the corresponding Schrodingers equation.where m is the reduced mass of the system given by:The corresponding wavefunctions are sketched belowThe energy levels are given byHarmonic OscillatorAlthough the above wavefunctions may appear sinusoidal they are bit more complexThey have more of a bell-shaped or gaussian appearance. Also note that for quantum number u=0 we have a finite energy known as zero point energy. The fundamental frequency of the oscillator,n0 is still given by classical value:However as before the energy values are quantized. The most interesting aspect of the Harmonic oscillator problem is the zero point energy. Which means a true quantum mechanical harmonic oscillator will continue to oscillate even at zero temperature!Hydrogen AtomAs we have seen previously, development of quantum mechanics was spurred by the observation of the spectrum of hydrogen. Now, we apply the Schrodinger equation to calculate the wavefunctions and energies of for this problem. In this problem, the potential energy is simply coulombic interaction. The final energies are: The corresponding wavefunction is bit more interesting. Unlike Bohr picture we find there are 2 additional quantum numbers associated with the azimuthal and polar angles f and q. (Recall that Bohr had initially proposed quantization of angular momentum). To describe the electron behavior in hydrogen atom problem, we need to specify three quantities, position (r) and two angles i.e. q and f.Hydrogen atom Wave functionsFor l=0 we have spherical symmetric wavefunctions, the s orbitals. For l=1, we have p orbitals; note the x,y,z dependence of the p orbitals in the table above. Physically, l quantum number can have values from 0 to n-1, and m quantum number can assume value like l, -l+1 .0.l-1,l, altogether 2l+1 values. The reason m is called, as the magnetic quantum number, is that only during the presence of magnetic field can we distinguish their separate values. In addition, when we consider the electron spin we have an additional spin quantum number. Thus, the hydrogen problem actually leads to definition of four quantum numbers. Nature of Hydrogen atom yNote although y has a maximu