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This gives an infinite number of cosine and sine functions that can be combined as a Fourier sum to satisfy a particular initial condition
$$V(X,0) = \sum_{i} (A_{i}\sin(\alpha_{i}X) + B_{i}\cos(\alpha_{i}X))$$
(15.23)
(Note that eð<sup>1</sup> <sup>þ</sup> 2Þ<sup>T</sup> ¼ 1 at T ¼ 0.) We can determine Ai and Bi b... | {
"Header 1": "Cable theory",
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At T=0 the sum cancels out the steady-state $\cosh(L-X)/\cosh(L)$ term leaving the voltage zero everywhere. As the exponentials decay, the terms in the sum gradually disappear and we arrive at the final steady state.
Equation (15.36) is plotted for L=2 and for X=0.1, 0.5, 1, and 2 (Fig. 15.4). This graph shows ho... | {
"Header 1": "Cable theory",
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Cable theory is clearly appropriate for axons and muscle fibers because they are shaped like cables, but what about cells with other geometries? Many neurons have extraordinarily complex dendrites (Fig. 15.5). Their extensive branching looks like a hopeless obstacle to mathematical modeling. However, in 1959 W. Rall ca... | {
"Header 1": "15.6 Branches and equivalent cylinder representations",
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The branch point, where X<sup>p</sup> ¼ L<sup>p</sup> and Xb1 ¼ Xb2 ¼ 0, provides two additional conditions. All three voltage functions must be equal at this point; Vp(Lp, T) ¼ Vb1(0, T) ¼ Vb2(0, T). So Vp(Lp) is simply Bpeð<sup>1</sup> <sup>þ</sup> 2Þ<sup>T</sup> because cos(0) ¼ 1 and sin(0) ¼ 0, and Ab1 and Ab2 a... | {
"Header 1": "15.6 Branches and equivalent cylinder representations",
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The tip of a patch pipette provides a direct electrical link from the amplifier to the interior of the cell. Experimentally, a voltage is imposed at the point denoted by V<sup>c</sup> in Fig. 15.7.
Before examining the complete model shown in Fig. 15.7 it is worth a brief comment on how this system performs when ther... | {
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The current is fitted to a sum of three exponentials, and -<sup>1</sup> and -2 can be used to determine L and <sup>m</sup> with Eqs. (15.30)–(15.32).
The charging transient also can be used to evaluate the validity of the equivalent cylinder representation. For example, Eq. (15.32) tells us that if -<sup>1</sup> > 9-... | {
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The plots show how a very brief transient signal is shaped as it spreads.
Figure 15.11 makes a number of important points about the passive spread of voltage signals. A proximal input, one near the recording site, rises almost instantly. It decays rapidly at first and slowly later on. A distal input, one far from the... | {
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Fig: 15:12: (a) Responses to a fast input ( ¼ 5), recorded at Z ¼ 0.01 in a cable of length L ¼ 2, with inputs at the indicated locations. Equation (15.84) was summed to 100 terms. The dotted curve represents the input ('(S) in Eq. (15.81)). (b) As in (a) but with ¼ 0.2.
![](_page_438... | {
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NEURON then integrated these 200 coupled equations, with the conductance of one of the compartments changing to simulate a synapse at the various sites indicated (input function was Eq. (15.81) with ¼ 5).
As in Figs. 15.11 and 15.12a, the synapses at more distant inputs produce responses at the cell body that are sma... | {
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Many types of cells, including neurons, muscle fibers, and endocrine cells, have the capacity to generate electrical impulses. These impulses, known as action potentials, play an important role in the regulation of cell function, and constitute a biological mechanism for the digitization of information. Action potentia... | {
"Header 1": "**Action potentials**",
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The delays to peak in (b) reflect the time for the action potential to propagate to those points.
then be depolarized sufficiently to open new Na<sup>þ</sup> channels and initiate a new action potential. In this way the action potential can propagate all the way down the cable to its fa... | {
"Header 1": "**Action potentials**",
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Section 16.5 will show how these ideas can be extended to understand propagation.
The quantitative description of Na<sup>+</sup> and K<sup>+</sup> channels introduced by Hodgkin and Huxley (1952) is still in wide use today. They fitted the separate components of Na<sup>+</sup> and K<sup>+</sup> current in Fig. 16.5 t... | {
"Header 1": "**Action potentials**",
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When h and n catch up with m, their processes of Na<sup>+</sup>
channel inactivation and K<sup>+</sup> channel activation, respectively, initiate the recovery of the action potential and the return of the voltage toward its resting value.
For any voltage, we can take the measured values of $\tau_m$ and $m_\infty... | {
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(16.9), set dV/dt = 0, and set m, h, and n equal to $m_{\infty}$ , $h_{\infty}$ , and n. The result is a steady-state current–voltage relation. Calling this voltage $V_{s-s}$ , we have
$$\begin{split} -I_{\text{stim}} &= (V_{\text{s-s}} - E_{\text{Na}})G_{\text{Na-max}} m_{\infty} (V_{\text{s-s}})^3 h_{\infty} (V_... | {
"Header 1": "**Action potentials**",
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The second term is the current through the membrane. To incorporate current through the channels into the cable equation, we replace this second term, $V/r_{\rm m}$ , with the sum of the membrane current terms from Eq. (16.9)
$$c_{m'} \frac{\partial V}{\partial t} = \frac{1}{r_{a}} \frac{\partial^{2} V}{\partial x^{... | {
"Header 1": "**Action potentials**",
"token_count": 2043,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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With myelin, a 10-*m*m diameter nerve fiber of a frog conducts at about 20 m s-<sup>1</sup> (Hodgkin, 1964), which is roughly three times faster than a 100-*m*m diameter axon in the squid (Fig. 16.9).
Myelin sheaths do not cover the entire axon surface. There are bare spots, called nodes of Ranvier, spaced at regular... | {
"Header 1": "**Action potentials**",
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This is not a significant advantage in a complex behavior involving sequential activation of many neurons, because the accumulated synaptic delays may consume much more time, say 50 ms. On the other hand, a 3-fold reduction in diameter allows an animal to put 9 times as many axons into the same space, providing 2<sup>9... | {
"Header 1": "**Action potentials**",
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There was a Na<sup>þ</sup> current with qualitatively similar behavior to the squid axon Na<sup>þ</sup> current, and two K<sup>þ</sup> currents. One of the K<sup>þ</sup> currents, IK, bore some resemblance to that of the squid axon, but the other, I<sup>A</sup> (the A-current), was very different. Figure 16.13 reproduc... | {
"Header 1": "**Action potentials**",
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(1.28)) is used to express the equilibrium voltage dependence of the $Ca^{2+}$ channel open probability, $m_{\infty}$ , as follows
$$m_{\infty} = \frac{1}{1 + e^{-V/7.5}} \tag{16.23}$$
The K<sup>+</sup> channels open at a more positive voltage, so the Boltzmann equation for $n_{\infty}$ , the K<sup>+</sup> chan... | {
"Header 1": "**Action potentials**",
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16.16.
The concept of n–V space provides a useful method of visualizing oscillations. Figure 16.17 plots n versus V together with the null-clines. The oscillating voltage for $I_{\rm stim}=80\,{\rm nA}$ settles into a closed loop that cycles endlessly in the counterclockwise direction. The direction is evident from... | {
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#### AI.I Taylor series
Any function can be approximated in the vicinity of a particular point by a tangent line through that point. This approximation takes the form
$$F(x + \delta x) \approx F(x) + \delta x \frac{dF(x)}{dx}$$
(A1.1)
This is illustrated graphically in Fig. A1.1. The deterioration of this approxi... | {
"Header 1": "Appendix I Expansions and series",
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It is easy to check by multiplying out the product that
$$(1-\alpha)(1+\alpha+\alpha^2+\alpha^3 \dots \alpha^n) = 1-\alpha^{n+1}$$
(A1.8)
The sum of a geometric series is the second factor on the left, so
$$\sum_{i=0}^{n} \alpha^{i} = \frac{1 - \alpha^{n+1}}{1 - \alpha} \tag{A1.9}$$
Differentiating this express... | {
"Header 1": "Appendix I Expansions and series",
"Header 2": "AI.3 Geometric series",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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#### A2.1 Linear transforms
Matrices simplify the mathematical analysis of problems with multiple linear equations and variables. For example, four linear equations with four unknowns take the form
$$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 + a_{14}x_4 = y_1 (A2.1a)$$
$$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4 = y_... | {
"Header 1": "Appendix I Expansions and series",
"Header 2": "Appendix 2 Matrix algebra",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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$$|\mathbf{A}\mathbf{B}| = |\mathbf{A}||\mathbf{B}| \tag{A2.12}$$
For $2 \times 2$ matrices
$$|\mathbf{A}||\mathbf{B}| = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} \begin{vmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{vmatrix} = (a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21})$... | {
"Header 1": "Appendix I Expansions and series",
"Header 2": "Appendix 2 Matrix algebra",
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The off-diagonal elements on the right-hand side are all zero because $\mathbf{x}_i^t \mathbf{x}_j = 0$ , and the diagonal elements are the eigenvalues, because $\mathbf{x}_i^2 = 1$ . So we have
$$\mathbf{X}^{\mathbf{t}}\mathbf{A}\mathbf{X} = \mathbf{\Lambda} \tag{A2.22}$$
where $\Lambda$ is a diagonal matrix w... | {
"Header 1": "Appendix I Expansions and series",
"Header 2": "Appendix 2 Matrix algebra",
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Fourier analysis enables one to express functions as sums of the trigonometric sine and cosine functions. We will first show how this works with an example. Take the periodic function that flips between +1 and -1 at regular intervals of $\pi$ (Fig. A3.1). This function is perfectly represented by the sum
$$F(x) = \... | {
"Header 1": "Appendix 3 Fourier analysis",
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Multiplying by -ix again gives
$$\phi\left(\frac{\mathrm{d}^2G}{\mathrm{d}s^2}\right) = -x^2\phi(G) \tag{A3.15}$$
#### Appendix 4
#### Gaussian integrals
The integral of a Gaussian function cannot be evaluated for arbitrary limits
$$\vartheta = \int_{a}^{b} e^{-\alpha x^{2}} x \tag{A4.1}$$
However, it can b... | {
"Header 1": "Appendix 3 Fourier analysis",
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The hyperbolic cosine is defined as
$$cosh(x) = \frac{e^x + e^{-x}}{2} \tag{A5.1}$$
The hyperbolic sine is
$$sinh(x) = \frac{e^x - e^{-x}}{2}$$
(A5.2)
It is easy to see that the two are interconverted by differentiation. The hyperbolic tangent is defined as the ratio of the two, just as the trigonometric tangen... | {
"Header 1": "Appendix 5 Hyperbolic functions",
"token_count": 419,
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In many problems everything revolves around a central point and the key variable is the distance, r, to this center. In these cases one uses polar or spherical coordinates, and writes an equation as a function of r. The standard operations in calculus must then be modified. When everything lies in one plane, then we go... | {
"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "Appendix 6 Polar and spherical coordinates",
"token_count": 974,
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- Abbott, A. J. and Nelsestuen, G. L. (1988). The collisional limit: an important consideration for membrane-associated enzymes and receptors. Faseb J., 2, 2858–2866.
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Biol., 5, 457–463.
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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and Hawkes, A. G. (1995). A Q-matrix cookbook. In Single-Channel Recording, ed. B. Sakmann and E. Neher. New York: Plenum, pp. 589–633.
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
and Redman, S. J. (1983). The synaptic current evoked in cat spinal motoneurons by impulse in single group Ia axons. J. Physiol., 342, 615–632.
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Physiol., 72, 409–442. [Cited in Chapter 14. A full treatment of single-file models and an excellent summary of their basic properties.]
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Sci., 98, 2958–2960. [Cited in Chapter 2. This presents a clear presentation of the oil-droplet versus jigsaw puzzle pictures of a protein interior.]
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"Header 1": "Appendix 5 Hyperbolic functions",
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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"Header 1": "Appendix 5 Hyperbolic functions",
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Acta, 1458, 88–103.
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
"token_count": 2027,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "References",
"token_count": 836,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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| absorbing boundary conditions 148, | helix continuation parameter 81 | axial resistivity 401 |
|---------------------------------------|-----------------------------------------|-----------------------------------------|
| 184, 198, 208, 213, 406 | alcohols 48 ... | {
"Header 1": "Appendix 5 Hyperbolic functions",
"Header 2": "Index",
"token_count": 17614,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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UNIVERSITY OF CALIFORNIA, DAVIS
**john D. Simon**
George B. Geller Professor of Chemistry DUKE UNIVERSITY
J
**c;Lf/.3 M** *t1'5*
**University Science Books**
,~ 55D Gate Five Road . ~'',.~Sausalito, CA 94965 ~~~....,:~ · Fax: (415) 332-5393
> Prod... | {
"Header 2": "**Donald A. McQuarrie**",
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- 1-1. Blackbody Radiation Could Not Be Explained by Classical Physics 2
- 1-2. Planck Used a Quantum Hypothesis to Derive the Blackbody Radiation Law 4
- 1-3. Einstein Explained the Photoelectric Effect with a Quantum Hypothesis 7
- 1-4. The Hydrogen Atomic Spectrum Consists of Several Series of Lines 10
- 1-5. The Ry... | {
"Header 2": "**CHAPTER 1** I The Dawn of the Quantum Theory",
"token_count": 231,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- **3-1.** The Schrödinger Equation Is the Equation for Finding the Wave Function of a Particle 73
- **3-2.** Classical-Mechanical Quantities Are Represented by Linear Operators in Ouantum Mechanics 75
- 3-3. The Schrödinger Equation Can Be Formulated As an Eigenvalue Problem 77
- **3-4.** Wave Function's Have a Probab... | {
"Header 2": "**MATHCHAPTER 8** I Probability and Statistics 63",
"Header 3": "**CHAPTER 3** / The Schrödinger Equation and a Particle In a Box 73",
"token_count": 250,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- **5-1.** A Harmonic Oscillator Obeys Hooke's Law 157
- **5-2.** The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule 161
- **5-3.** The Harmonic-Oscillator Approximation Results from the Expansion of an Internuclear Potential Around Its Minimum 163
- \* 5-4. Th... | {
"Header 2": "CHAPTER 5 / The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models 157",
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- 6-1. The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly 191
- 6-2. The Wave Functions of a Rigid Rotator Are Called Spherical Harmonics 193
- 6-3. The Precise Values of the Three Components of Angular Momentum Cannot Be Measured Simultaneously 200
- 6-4. Hydrogen Atomic Orbitals Depend upon Three Qu... | {
"Header 2": "CHAPTER 5 / The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models 157",
"Header 3": "CHAPTER 6 I The Hydrogen Atom 191",
"token_count": 213,
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- <sup>~</sup>7-1. The Variational Method Provides an Upper Bound to the Ground-State Energy of a System 241
- 7-2. A Trial Function That Depends Linearly on the Variational Parameters Leads to a Secular Determinant 249
- 7-3. Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Paramet... | {
"Header 2": "CHAPTER 5 / The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models 157",
"Header 3": "CHAPTER 7 I Approximation Methods 241",
"token_count": 910,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- 12-1. The Exploitation of the Symmetry of a Molecule Can Be Used to Significantly Simplify Numerical Calculations 453
- 12-2. The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements 455
- 12-3. The Symmetry Operations of a Molecule Form a Group 460
- 12-4. Symmetry Operations Can Be Represented by Ma... | {
"Header 2": "**CHAPTER 11** I Computational Quantum Chemistry 411",
"Header 3": "CHAPTER 12 I Group Theory: The Exploitation of Symmetry 453",
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} |
- 13-1. Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different Molecular Processes 495
- 13-2. Rotational Transitions Accompany Vibrational Transitions 497
- 13-3. Vibration-Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotatio... | {
"Header 2": "**CHAPTER 11** I Computational Quantum Chemistry 411",
"Header 3": "*(.§;* CHAPTER 13 I Molecular Spectroscopy 495",
"token_count": 405,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- 14-1. Nuclei Have Intrinsic Spin Angular Momenta 548
- 14-2. Magnetic Moments Interact with Magnetic Fields 550
- 14-3. Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz 554
- 14-4. The Magnetic Field Acting upon Nuclei in Molecules Is Shielded 556
- 14-5. Chemical Shifts Depend upon the Chem... | {
"Header 2": "**CHAPTER 11** I Computational Quantum Chemistry 411",
"Header 3": "CHAPTER 14 I Nuclear Magnetic Resonance Spectroscopy 547",
"token_count": 243,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- **15-1.** Electronically Excited Molecules Can Relax by a Number of Processes 592
- **15-2.** The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms Can Be Modeled by Rate Equations 595
- 15-3. A Two-Level System Cannot Achieve a Population Inversion 601
- **15-4.** Population Inversion Can ... | {
"Header 2": "**CHAPTER 11** I Computational Quantum Chemistry 411",
"Header 3": "CHAPTER 15 / Lasers, Laser Spectroscopy, and Photochemistry 591",
"token_count": 256,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- **16-1.** All Gases Behave Ideally If They Are Sufficiently Dilute 637
- **16-2.** The van der Waals Equation and the Redlich–Kwong Equation Are Examples of Two-Parameter Equations of State 642
- **16-3.** A Cubic Equation of State Can Describe Both the Gaseous and Liquid States 648
- **16-4.** The van der Waals Equa... | {
"Header 2": "**CHAPTER 11** I Computational Quantum Chemistry 411",
"Header 3": "**CHAPTER 16** / The Properties of Gases 637",
"token_count": 233,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **17-1.** The Boltzmann Factor Is One of the Most Important Quantities in the Physical Sciences 694
- **17-2.** The Probability That a System in an Ensemble Is in the State j with Energy $E_j(N, V)$ Is Proportional to $e^{-E_j(N,V)/k_BT}$ 696
- 17-3. We Postulate That the Average Ensemble Energy Is Equal to the O... | {
"Header 2": "CHAPTER 17 The Boltzmann Factor and Partition Functions 693",
"token_count": 615,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **19-1.** A Common Type of Work is Pressure Volume Work 766
- **19-2.** Work and Heat Are Not State Functions, but Energy Is a State Function 769
- **19-3.** The First Law of Thermodynamics Says the Energy Is a State Function 773
- **19-4.** An Adiabatic Process Is a Process in Which No Energy as Heat Is Transferred ... | {
"Header 2": "CHAPTER 17 The Boltzmann Factor and Partition Functions 693",
"Header 3": "**CHAPTER 19** / The First Law of Thermodynamics 765",
"token_count": 351,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **20-1.** The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process 817
- **20-2.** Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Disorder 819
- **20-3.** Unlike $q_{rev}$ Entropy Is a State Function 821
- 20-4. The Second Law of Thermodynamics Sta... | {
"Header 2": "**CHAPTER 20** / Entropy and the Second Law of Thermodynamics 817",
"token_count": 297,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- **21-1.** Entropy Increases with Increasing Temperature 853
- **21-2.** The Third Law of Thermodynamics Says That the Entropy of a Perfect Crystal Is Zero at 0 K 855
- **21-3.** $\Delta_{\text{trs}}S = \Delta_{\text{trs}}H/T_{\text{trs}}$ at a Phase Transition 857
- **21-4.** The Third Law of Thermodynamics Asserts... | {
"Header 2": "CHAPTER 21 / Entropy and the Third Law of Thermodynamics 853",
"token_count": 291,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **22-1.** The Sign of the Helmholtz Energy Change Determines the Direction of a Spontaneous Process in a System at Constant Volume and Temperature 881
- **22-2.** The Gibbs Energy Determines the Direction of a Spontaneous Process for a System at Constant Pressure and Temperature 884
- 22-3. Maxwell Relations Provide ... | {
"Header 2": "CHAPTER 22 / Helmholtz and Gibbs Energies 881",
"token_count": 381,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- 24-1. Chemical Equilibrium Results when the Gibbs Energy Is a Minimum with Respect to the Extent of Reaction 963
- 24-2. An Equilibrium Constant Is a Function of Temperature Only 967
- 24-3. Standard Gibbs Energies of Formation Can Be Used to Calculate Equilibrium Constants 970
- 24-4. A Plot of the Gibbs Energy of a... | {
"Header 2": "CHAPTER 22 / Helmholtz and Gibbs Energies 881",
"Header 3": "CHAPTER 24 I Chemical Equilibrium 963",
"token_count": 325,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- 25-1. The Average Translational Kinetic Energy of the Molecules in a Gas Is Directly Proportional to the Kelvin Temperature 1011
- 25-2. The Distribution of the Components of Molecular Speeds Are Described by a Gaussian Distribution 1016
- 25-3. The Distribution of Molecular Speeds Is Given by the Maxwell-Boltzmann D... | {
"Header 2": "CHAPTER 25 I TheKineticTheoryofGases 1011",
"token_count": 794,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **28-1.** The Rate of a Bimolecular Gas-Phase Reaction Can Be Calculated Using Hard-Sphere Collision Theory and an Energy-Dependent Reaction Cross Section 1139
- 28-2. A Reaction Cross Section Depends upon the Impact Parameter 1144
- **28-3.** The Rate Constant for a Gas-Phase Chemical Reaction May Depend on the Orie... | {
"Header 2": "CHAPTER 28 / Gas-Phase Reaction Dynamics 1139",
"token_count": 352,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**29-1.** The Unit Cell Is the Fundamental Building Block of a Crystal 1181
- **29-2.** The Orientation of a Lattice Plane Is Described by Its Miller Indices 1181
- **29-3.** The Spacing Between Lattice Planes Can Be Determined from X-Ray Diffraction Measurements 1191
- **29-4.** The Total Scattering Intensity Is Rel... | {
"Header 2": "**CHAPTER 29** / Solids and Surface Chemistry 1181",
"token_count": 371,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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You are about to begin your study of physical chemistry. You may have been told that physical chemistry is the most difficult chemistry course that you will take, or you may have even seen the bumper sticker that says "Honk if you passed P Chern." The anxiety that some students bring to their physical chemistry course ... | {
"Header 2": "**Preface**",
"Header 3": "**To the Student**",
"token_count": 2032,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Some instructors have raised the question about whether these omitted topics are on the GRE exam and whether their students would be disadvantaged. The answer to this question is "No." In fact, one of us was recently on the GRE Chemistry Board for eight years and was Chair for the last two of them. More than ten year... | {
"Header 2": "**Preface**",
"Header 3": "**To the Student**",
"token_count": 711,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Many people have contributed to the writing of this book. We thank our colleagues, Paul Barbara, James T. Hynes, Veronica Vaida, John Crowell, Andy Kummel, Robert Continetti, Amit Sinha, John Weare, Kim Baldridge, Jack Kyte, and Bill Trogler for stimulating discussions on the topics that should be included in a modem p... | {
"Header 2": "**Acknowledgments**",
"token_count": 375,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
A MOLECULAR APPROACH
**Max Planck** was born in Kiel, Germany (then Prussia) on April23, 1858, and died in 1948. He showed early talent in both music and science. He received his Ph.D. in theoretical physics in 1879 at the University of Munich for his dissertation on the second law of t... | {
"Header 2": "PHYSICAL CHEMISTRY",
"token_count": 364,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Toward the end of the nineteenth century, many scientists believed that all the fundamental discoveries of science had been made and little remained but to clear up a few minor problems and to improve experimental methods to measure physical results to a greater number of decimal places. This attitude was somewhat just... | {
"Header 2": "**The Dawn of the Quantum Theory**",
"token_count": 895,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The series of experiments that revolutionized the concepts of physics had to do with the radiation given off by material bodies when they are heated. We all know, for instance, that when the burner of an electric stove is heated, it first turns a dull red and progressively becomes redder as the temperature increases. W... | {
"Header 2": "1-1. Blackbody Radiation Could Not Be Explained by Classical Physics",
"token_count": 973,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The first person to offer a successful explanation of blackbody radiation was the German physicist Max Planck in 1900. Like Rayleigh and Jeans before him, Planck assumed that the radiation emitted by the blackbody was caused by the oscillations of the electrons in the constituent particles of the material body. These e... | {
"Header 2": "**1–2.** Planck Used a Quantum Hypothesis to Derive the Blackbody Radiation Law",
"token_count": 1963,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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In 1886 and 1887, while carrying out the experiments that supported Maxwell's theory of the electromagnetic nature of light, the German physicist Heinrich Hertz discovered that ultraviolet light causes electrons to be emitted from a metallic surface. The ejection of electrons from the surface of a metal by radiation is... | {
"Header 2": "**1-3.** Einstein Explained the Photoelectric Effect with a Quantum Hypothesis",
"token_count": 1890,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
For some time scientists had known that every atom, when subjected to high temperatures or an electrical discharge, emits electromagnetic radiation of characteristic frequencies. In other words, each atom has a characteristic emission spectrum. Because the emission spectra of atoms consist of only certain discrete freq... | {
"Header 2": "**1–4.** The Hydrogen Atomic Spectrum Consists of Several Series of Lines",
"token_count": 1365,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The Swiss spectroscopist Johannes Rydberg accounted for all the lines in the hydrogen atomic spectrum by generalizing the Balmer formula to
$$\tilde{v} = \frac{1}{\lambda} = 109680 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \text{cm}^{-1} \quad (n_2 > n_1)$$
(1.10)
where both $n_1$ and $n_2$ are integers ... | {
"Header 2": "**1–5.** The Rydberg Formula Accounts for All the Lines in the Hydrogen Atomic Spectrum",
"token_count": 1050,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Although we have an intriguing partial insight into the electronic structure of atoms, something is missing. To explore this further, let us go back to a discussion of the nature of light.
Scientists have always had trouble describing the nature of light. In many experiments light shows a definite wavelike character,... | {
"Header 2": "**1–5.** The Rydberg Formula Accounts for All the Lines in the Hydrogen Atomic Spectrum",
"Header 3": "1-6. Louis de Broglie Postulated That Matter Has Wavelike Properties",
"token_count": 1243,
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When a beam of X rays is directed at a crystalline substance, the beam is scattered in a definite manner characteristic of the atomic structure of the crystalline substance. This phenomenon is called *X-ray diffraction* and occurs because the interatomic spacings in
**TABlE 1.2** The de Broglie wavelengths of variou... | {
"Header 2": "**1-7.** de Broglie Waves Are Observed Experimentally",
"token_count": 1129,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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In 1911, the Danish physicist Niels Bohr presented a theory of the hydrogen atom that gave a beautifully simple explanation of the hydrogen atomic spectrum. We present here a brief discussion of the Bohr theory.
According to the nuclear model of the atom, the hydrogen atom can be pictured as a central, rather massive... | {
"Header 2": "**1–8.** The Bohr Theory of the Hydrogen Atom Can Be Used to Derive the Rydberg Formula",
"token_count": 1950,
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Bohr assumed that the observed spectrum of the hydrogen atom is due to transitions from one allowed energy state to another, and using Equation 1.22, he predicted that the allowed energy differences are given by
$$\Delta E = \frac{m_e e^4}{8\varepsilon_0^2 h^2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = hv \t... | {
"Header 2": "**1–8.** The Bohr Theory of the Hydrogen Atom Can Be Used to Derive the Rydberg Formula",
"token_count": 489,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Using the values of the physical constants given inside the front cover of this book, calculate *R* and compare the result to its experimental value, 109 677.6 cm- <sup>1</sup> • 00
SOLUTION:
$$R_{\infty} = \frac{(9.10939 \times 10^{-31} \text{ kg})(1.602177 \times 10^{-19} \text{ C})^4}{(8)(8.85419 \times 10^{-12}... | {
"Header 2": "**1–8.** The Bohr Theory of the Hydrogen Atom Can Be Used to Derive the Rydberg Formula",
"Header 3": "**EXAMPLE 1-8**",
"token_count": 651,
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We now know that we must consider light and matter as having the characteristics of both waves and particles. Let's consider a measurement of the position of an electron. If we wish to locate the electron within a distance ~x, then we must use a measuring device that has a spatial resolution less than ~x. One way to ac... | {
"Header 2": "1-9. The Heisenberg Uncertainty Principle States That the Position and the Momentum of a Particle Cannot Be Specified Simultaneously with Unlimited Precision",
"token_count": 1263,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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**1-1.** Radiation in the ultraviolet region of the electromagnetic spectrum is usually described in terms of wavelength, $\lambda$ , and is given in nanometers ( $10^{-9}$ m). Calculate the values of $\nu$ , $\tilde{\nu}$ , and E for ultraviolet radiation with $\lambda=200$ nm and compare your results with those... | {
"Header 2": "1-9. The Heisenberg Uncertainty Principle States That the Position and the Momentum of a Particle Cannot Be Specified Simultaneously with Unlimited Precision",
"Header 3": "**Problems**",
"token_count": 1871,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Identify the spectral regions to which these wavelengths correspond.
- **1-24.** Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.
- **1-25.** Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of 100 eV, (b) a proton with a kinetic ener... | {
"Header 2": "1-9. The Heisenberg Uncertainty Principle States That the Position and the Momentum of a Particle Cannot Be Specified Simultaneously with Unlimited Precision",
"Header 3": "**Problems**",
"token_count": 2002,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Throughout physical chemistry, we frequently use complex numbers. In this mathchapter, we review some of the properties of complex numbers. Recall that complex numbers involve the imaginary unit, i, which is defined to be the square root of -1:
$$i = \sqrt{-1} \tag{A.1}$$
or
$$i^2 = -1 \tag{A.2}$$
Complex numbe... | {
"Header 2": "COMPLEX NUMBERS",
"token_count": 1905,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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A-1. Find the real and imaginary parts of the following quantities:
**a.**
$$(2-i)^3$$
**h.**
$$e^{\pi i/2}$$
**c.**
$$e^{-2+i\pi/2}$$
**d.**
$$(\sqrt{2}+2i)e^{-i\pi/2}$$
**A-2.** If z = x + 2iy, then find
a.
$$Re(z^*)$$
**b.**
$$Re(z^2)$$
c.
$$Im(z^2)$$
**d.**
$$Re(zz^*)$$
e.
$$Im(zz^*)$$
**A-3.*... | {
"Header 2": "**Problems**",
"token_count": 2024,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In 1925, Erwin Schrodinger and Werner Heisenberg independently formulated a general quantum theory. At first sight, the two methods appeared different because Heisenberg's method is formulated in terms of matrices, whereas Schr6dinger' s method is formulated in terms of partial differential equations. Just a year later... | {
"Header 2": "**The Classical Wave Equation**",
"token_count": 336,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Consider a uniform string stretched between two fixed points as shown in Figure 2.1. The maximum displacement of the string from its equilibrium horizontal position is 3 9
FIGURE 2.1 A vibrating string whose ends are fixed at 0 and l. The amplitude of the vibration at position *x* and t... | {
"Header 2": "**2-1.** The One-Dimensional Wave Equation Describes the Motion of a Vibrating String",
"token_count": 458,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The classical wave equation, as well as the Schrodinger equation and many other partial differential equations that arise in physical chemistry, can often be solved by a method called *separation of variables.* We shall use the problem of a vibrating string to illustrate this method.
The key step in the method of sep... | {
"Header 2": "2-2. The Wave Equation Can Be Solved by the Method of Separation of Variables",
"token_count": 2007,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Now let's consider the case where K < 0 in Equations 2.8 and 2.9. In this case, *a* will be imaginary. As a concrete example, consider the differential equation
$$\frac{d^2y}{dx^2} + y(x) = 0 (2.16)$$
which is essentially Equation 2. 8 with K = -1. If we let *y* (x) = *eax,* we have
$$\left(\alpha^2 + 1\right) y(... | {
"Header 2": "2-2. The Wave Equation Can Be Solved by the Method of Separation of Variables",
"Header 3": "2-3. Some Differential Equations Have Oscillatory Solutions",
"token_count": 1060,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Let us assess where we are now. We have obtained Equations 2.8 and 2.9 by applying the method of separation of variables to the wave equation. We have already shown that if the separation constant K is zero, then only a trivial solution results. Now let's assume that K is positive. To this end, write K as $\beta^2$ , ... | {
"Header 2": "**2–4.** The General Solution to the Wave Equation Is a Superposition of Normal Modes",
"token_count": 2023,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The generalization of Equation 2.1 to two dimensions is
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$$
(2.28)
An illustration of how two standing waves can combine to give a traveling wave. In both parts, time... | {
"Header 2": "**2–5.** A Vibrating Membrane Is Described by a Two-Dimensional Wave Equation",
"token_count": 2002,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The case in which a = b is an
#### FIGURE 2.6
The first few normal modes of a rectangular membrane with shaded and clear sections having opposite sinusoidal displacements as indicated.
**F I CURE** 2.7
The normal modes of a square membrane, illustra... | {
"Header 2": "**2–5.** A Vibrating Membrane Is Described by a Two-Dimensional Wave Equation",
"token_count": 558,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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**2-1.** Find the general solutions to the following differential equations.
**a.**
$$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 3y = 0$$
**b.** $\frac{d^2y}{dx^2} + 6\frac{dy}{dx} = 0$ **c.** $\frac{dy}{dx} + 3y = 0$ **d.** $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} - y = 0$ **e.** $\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y... | {
"Header 2": "**Problems**",
"token_count": 2015,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**2-9.** We will see in Chapter 3 that the Schrödinger equation for a particle of mass *m* that is constrained to move freely along a line between 0 and *a* is
$$\frac{d^2\psi}{dx^2} + \left(\frac{8\pi^2 mE}{h^2}\right)\psi(x) = 0$$
with the boundary condition
$$\psi(0) = \psi(a) = 0$$
In this equation, E is ... | {
"Header 2": "**Problems**",
"token_count": 1988,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Show that for small angles, Newton's equation is
$$ml\frac{d^2\theta}{dt^2} + \lambda l\frac{d\theta}{dt} + mg\theta = 0$$
Show that there is no harmonic motion if
$$\lambda^2 > \frac{4m^2g}{l}$$
Does it make physical sense that the medium can be so viscous that the pendulum undergoes no harmonic motion?
![](... | {
"Header 2": "**Problems**",
"token_count": 897,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In many of the following chapters, we will deal with probability distributions, average values, and standard deviations. Consequently, we take a few pages here to discuss some basic ideas of probability and show how to calculate average quantities in general.
Consider some experiment, such as the tossing of a coin or... | {
"Header 2": "PROBABILITY AND STATISTICS",
"token_count": 464,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
| <i>x</i> | p(x) |
|----------|------|
| 1 | 0.20 |
| 3 | 0.25 |
| 4 | 0.55 |
Calculate the average value of x.
SOLUTION: Using Equation B.4, we have
$$\langle x \rangle = (1)(0.20) + (3)(0.25) + (4)(0.55) = 3.15$$
It is helpful to interpret a probability distribution like $p_j$ as a dis... | {
"Header 2": "**EXAMPLE B-1** Suppose we are given the following data:",
"token_count": 1794,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
SOLUTION: Because p(x) must be normalized,
$$\int_{a}^{b} p(x)dx = 1 = A \int_{a}^{b} dx = A(b - a)$$
Therefore, A = 1/(b - a) and
$$p(x) = \frac{1}{b-a} \qquad a \le x \le b$$
= 0 otherwise
The mean of x is given by
$$\langle x \rangle = \int_a^b x p(x) dx = \frac{1}{b-a} \int_a^b x dx$$
$$= \frac{b^2 - a^... | {
"Header 2": "**EXAMPLE B-1** Suppose we are given the following data:",
"token_count": 1856,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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**B-1.** Consider a particle to be constrained to lie along a one-dimensional segment 0 to a. We will learn in the next chapter that the probability that the particle is found to lie between x and x + dx is given by
$$p(x)dx = \frac{2}{a}\sin^2\frac{n\pi x}{a}dx$$
where $n = 1, 2, 3, \ldots$ First show that p(x) ... | {
"Header 2": "**EXAMPLE B-1** Suppose we are given the following data:",
"Header 3": "**Problems**",
"token_count": 1339,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The Schrodinger equation is our fundamental equation of quantum mechanics. The solutions to the Schrodinger equation are called *wave functions.* We will see that a wave function gives a complete description of any system. In this chapter, we present and discuss the version of the Schrodinger equation that does not con... | {
"Header 2": "**The Schrodinger Equation and a Particle In a Box**",
"token_count": 251,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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We cannot derive the Schrodinger equation any more than we can derive Newton's laws, and Newton's second law, f = *ma,* in particular. We shall regard the Schrodinger equation to be a fundamental postulate, or axiom, of quantum mechanics, just as Newton's laws are fundamental postulates of classical mechanics. Even tho... | {
"Header 2": "**The Schrodinger Equation and a Particle In a Box**",
"Header 3": "**3-1.** The Schrodinger Equation Is the Equation for Finding the Wave Function of a Particle",
"token_count": 1067,
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An *operator* is a symbol that tells you to do something to whatever follows the symbol. For example, we can consider *dy* I *dx* to be the *d* I *dx* operator operating on the function *y(x).* Some other examples are SQR (square what follows), J0 1 (integrate from 0 to 1), 3 (multiply by 3), and *a1ay* (partial deriva... | {
"Header 2": "**3-2.** Classical-Mechanical Quantities Are Represented by Linear Operators in Quantum Mechanics",
"token_count": 1262,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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A problem that we will frequently encounter in physical chemistry is the following: Given $\hat{A}$ , find a function $\phi(x)$ and a constant a such that
$$\hat{A}\phi(x) = a\phi(x) \tag{3.10}$$
Note that the result of operating on the function $\phi(x)$ by $\hat{A}$ is simply to give $\phi(x)$ back again... | {
"Header 2": "**3–3.** The Schrödinger Equation Can Be Formulated as an Eigenvalue Problem",
"token_count": 1766,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In this section, we will study the case of a free particle of mass *m* constrained to lie along the x-axis between *x* = 0 and *x* = *a.* This case is called the *problem of a particle in a one-dimensional box* (cf. Figure 3.1). It is mathematically a fairly simple problem, so we can study the solutions in great detail... | {
"Header 2": "**3–3.** The Schrödinger Equation Can Be Formulated as an Eigenvalue Problem",
"Header 3": "**3-4.** Wave Functions Have a Probabilistic Interpretation",
"token_count": 818,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
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