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Quantum Mechanics 187 5.11 HARMONIC OSCILLATOR Its energy levels are evenly spaced Harmonic motion takes place when a system of ...
x 0 (5.62) There are various ways to write the solution to Eq. (5.62). A common one is xAcos (2t ) (5.63) where (5 ...
Even before we make a detailed calculation we can anticipate three quantum- mechanical modifications to this classical picture: ...
The energy of a harmonic oscillator is thus quantized in steps of h. We note that when n 0, Zero-point energy E 0 ^12 h (5.7 ...
where it moves slowly, and least near the equilibrium position (x0), where it moves rapidly. Exactly the opposite behavior occu ...
Example 5.7 Find the expectation value x for the first two states of a harmonic oscillator. Solution The general formula for x i ...
Appendix to Chapter 5 The Tunnel Effect W e consider the situation that was shown in Fig. 5.9 of a particle of energy EUthat ap ...
is the flux of particles that arrive at the barrier. That is, Sis the number of particles per second that arrive there. At x 0 t ...
Applying the Boundary Conditions In order to calculate the transmission probability Twe have to apply the appropriate boundary c ...
Let us assume that the potential barrier Uis high relative to the energy Eof the incident particles. If this is the case, then k ...
Exercises 197 EXERCISES Press on, and faith will catch up with you. — Jean D’Alembert 5.1 Quantum Mechanics Which of the wave f ...
5.7 Schrödinger’s Equation: Steady-State Form Obtain Schrödinger’s steady-state equation from Eq. (3.5) with the help of de Bro ...
amplitude such that its bob rises a maximum of 1.00 mm above its equilibrium position. What is the corresponding quantum number? ...
200 CHAPTER 6 Quantum Theory of the Hydrogen Atom 6.1 SCHRÖDINGER’S EQUATION FOR THE HYDROGEN ATOM Symmetry suggests spherical p ...
T he first problem that Schrödinger tackled with his new wave equation was that of the hydrogen atom. He found the mathematics h ...
angle between the projection of the radius vector in the xy plane and the xaxis, measured in the direction shown azimuth ang ...
Quantum Theory of the Hydrogen Atom 203 E 0 (6.6) e^2 4 0 r 2 mr^2 sin^2 ^2 nevertheless free to move in three ...
The third term of Eq. (6.6) is a function of azimuth angle only, whereas the other terms are functions of rand only. Let us re ...
We have therefore accomplished our task of simplifying Schrödinger’s equation for the hydrogen atom, which began as a partial di ...
Hence we may tabulate the three quantum numbers n,l, and mtogether with their permissible values as follows: Principal quantum n ...
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