BioPHYSICAL chemistry
Properties of the general solution The solutions to the Schrödinger equation for the hydrogen atom should in principle allow us ...
than one nucleus? For the hydrogen atom, the solutions are given by the product of the three terms, yielding the solutions liste ...
The solutions can be written in terms of the individual solutions that we have found and the three quantum numbers: ψn,l,ml(r,θ, ...
So, the total angular momentum is quantized by land the projection of the angular momentum along the zdirection is quantized by ...
We can plot the radial dependence of these wavefunctions and see that the higher orbitals have a more complex distribution with ...
a certain point in space in ψ*ψdτthat exponentially decreases as the radius increases (Figure 12.3). We normally want to know th ...
This can be solved by using the product rule: (12.22) letting u=x^2 du= 2 xdx d 9 =e−^2 xdx 9 =−e−^2 x/2 (12.23) and another sub ...
All of these distances, the most probable radius (a 0 ), the average value (3/2a 0 ), and the radius at which a sphere encloses ...
This reduces to: (12.32) Using the relationships x=rsinθcosφ y=rsinθsinφ (12.33) e±iφ=cosφ±isinφ it is easy to show that the p o ...
Transitions For the hydrogen atom, the electron will reside in the 1s orbital. As was true for the other applications, the elect ...
In general, for absorption, an electron in an orbital with the angular momentum l must make a transition to an orbital with an a ...
fossil-fuel economy, a number of technical aspects must be improved: for instance, the cost of the fuel cells must be significan ...
are considerably higher, typically 0.6 V or more on an unmodified surface (both overpotentials can be a serious consideration on ...
Spin From Schrödinger’s equation for the hydrogen atom we obtained three indices, n,l, and ml, that represent the quantization o ...
determine the total spin and zprojection of spin accord- ing to the relationships: (12.40) Electrons, protons, and neutrons all ...
CHAPTER 12 THE HYDROGEN ATOM 259 which can be written for the time-independent case (eqn 9.39) as: (db12.46) The classical laws ...
260 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY the energy of each particle corresponding to their masses (technically the rest ma ...
Using the hydrogen atom solution (eqn 12.3) with a nuclear chargeZ the energy of an electron is: (12.41) The value of the energy ...
idea of this approach is to assume that the second term is much smaller than the first term. This means that we can write for th ...
CHAPTER 12 THE HYDROGEN ATOM 263 This is first solved at zeroth order; that is, neglecting the interactions between electrons. T ...
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