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The general solution of Equation 3.17 is (see Example 2–4)
$$\psi(x) = A\cos kx + B\sin kx$$
with
$$k = \frac{(2mE)^{1/2}}{\hbar} = \frac{2\pi (2mE)^{1/2}}{h}$$
(3.18)
The first boundary condition requires that $\psi(0) = 0$ , which implies immediately that A = 0 because $\cos(0) = 1$ and $\sin(0) = 0$ . Th... | {
"Header 2": "**3–3.** The Schrödinger Equation Can Be Formulated as an Eigenvalue Problem",
"Header 3": "**3–5.** The Energy of a Particle in a Box Is Quantized",
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Because $\psi^*(x)\psi(x)dx$ is the probability of finding the particle between x and x + dx, the probability of finding the particle within the interval $x_1 \le x \le x_2$ is
$$Prob(x_1 \le x \le x_2) = \int_{x_1}^{x_2} \psi^*(x) \psi(x) dx$$
(3.28)
#### EXAMPLE 3-6
Calculate the probability that a partic... | {
"Header 2": "**3–3.** The Schrödinger Equation Can Be Formulated as an Eigenvalue Problem",
"Header 3": "**3–5.** The Energy of a Particle in a Box Is Quantized",
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We can use the probability distribution $\psi_n^*(x)\psi_n(x)$ to calculate averages and standard deviations (MathChapter B) of various physical quantities such as position and momentum. Using the example of a particle in a box, we see that
$$\psi_n^*(x)\psi_n(x) dx = \frac{2}{a} \sin^2 \frac{n\pi x}{a} dx \qquad 0... | {
"Header 2": "**3–3.** The Schrödinger Equation Can Be Formulated as an Eigenvalue Problem",
"Header 3": "3-7. The Average Momentum of a Particle in a Box Is Zero",
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Now let's calculate the variance of the momentum, $\sigma_p^2 = \langle p^2 \rangle - \langle p \rangle^2$ , of a particle in a box. To calculate $\langle p^2 \rangle$ , we use
$$\langle p^2 \rangle = \int \psi_n^*(x) \hat{P}_x^2 \psi_n(x) dx \tag{3.39}$$
and remember that $\hat{P}_x^2$ means apply $\hat{P}_x$... | {
"Header 2": "**3–8.** The Uncertainty Principle Says That $\\sigma_p \\sigma_x > \\hbar/2$",
"token_count": 1223,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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The simplest three-dimensional quantum-mechanical system is the three-dimensional version of a particle in a box. In this case, the particle is confined to lie within a rectangular parallelepiped with sides of lengths *a, b,* and *c* (Figure 3.5). The Schrodinger equation for this system is the three-dimensional extens... | {
"Header 2": "**3–8.** The Uncertainty Principle Says That $\\sigma_p \\sigma_x > \\hbar/2$",
"Header 3": "3-9. The Problem of a Particle in a Three-Dimensional Box Is a Simple Extension of the One-Dimensional Case",
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The average position is given by
$$\langle \mathbf{r} \rangle = \int_0^a dx \int_0^b dy \int_0^c dz \psi^*(x, y, z) \hat{\mathbf{R}} \psi(x, y, z)$$
$$= \mathbf{i} \langle x \rangle + \mathbf{j} \langle y \rangle + \mathbf{k} \langle z \rangle$$
Let's evaluate $\langle x \rangle$ first. Using Equation 3.55, we ha... | {
"Header 2": "**3–8.** The Uncertainty Principle Says That $\\sigma_p \\sigma_x > \\hbar/2$",
"Header 3": "3-9. The Problem of a Particle in a Three-Dimensional Box Is a Simple Extension of the One-Dimensional Case",
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**3-1.** Evaluate $g = \hat{A}f$ , where $\hat{A}$ and f are given below:
$$\begin{array}{cccccccccccccccccccccccccccccccccccc$$
- 3-2. Determine whether the following operators are linear or nonlinear:
- **a.** $\hat{A}f(x) = SQRf(x)$ [square f(x)]
- **b.** $\hat{A}f(x) = f^*(x)$ [form the complex conjugate... | {
"Header 2": "**Problems**",
"token_count": 2007,
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- **3-19.** In going from Equation 3.34 to 3.35, we multiplied Equation 3.34 from the left by $\psi^*(x)$ and then integrated over all values of x to obtain Equation 3.35. Does it make any difference whether we multiplied from the left or the right?
- **3-20.** Calculate $\langle x \rangle$ and $\langle x^2 \ran... | {
"Header 2": "**Problems**",
"token_count": 1987,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Using this equation, show that
$$K_n = \frac{\pi^2 v^2 n^2 \rho}{4l} D_n^2 \sin^2(\omega_n t + \phi_n)$$
and
$$V_n = \frac{\pi^2 n^2 T}{4l} D_n^2 \cos^2(\omega_n t + \phi_n)$$
Using the fact that $v = (T/\rho)^{1/2}$ , show that
$$E_n = \frac{\pi^2 v^2 n^2 \rho}{4l} D_n^2$$
Note that the total energy, or i... | {
"Header 2": "**Problems**",
"token_count": 2000,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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A vector is a quantity that has both magnitude and direction. Examples of vectors are position, force, velocity, and momentum. We specify the position of something, for example, by giving not only its distance from a certain point but also its direction from that point. We often represent a vector by an arrow, where th... | {
"Header 2": "**VECTORS**",
"token_count": 1959,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
SOLUTION: Equation C.9 with A = B gives
$$\mathbf{A} \cdot \mathbf{A} = A_x^2 + A_y^2 + A_z^2 = |\mathbf{A}|^2$$
Therefore,
$$|\mathbf{A}| = (\mathbf{A} \cdot \mathbf{A})^{1/2} = (4+1+9)^{1/2} = \sqrt{14}$$
#### EXAMPLE C-3
Find the angle between the two vectors $\mathbf{A} = \mathbf{i} + 3\mathbf{j} - \ma... | {
"Header 2": "**VECTORS**",
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Another example that involves a cross product is the equation that gives the force $\mathbf{F}$ on a particle of charge q moving with velocity $\mathbf{v}$ through a magnetic field $\mathbf{B}$ :
$$\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$$
Note that the force is perpendicular to $\mathbf{v}$ , and so t... | {
"Header 2": "**VECTORS**",
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"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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- C-1. Find the length of the vector v = 2i j + 3k.
- C-2. Find the length of the vector r = xi + *y* j and of the vector r = xi+ *y* j + zk.
- C-3. Prove that A· B = 0 if A and B are perpendicular to each other. Two vectors that are perpendicular to each other are said to be orthogonal.
- C-4. Show that the vectors A ... | {
"Header 2": "**VECTORS**",
"Header 3": "Problems",
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Classical mechanics deals with quantities called *dynamical variables*, such as position, momentum, angular momentum, and energy. A measurable dynamical variable is called an *observable*. The classical-mechanical state of a particle at any particular time is specified completely by the three position coordinates (x, y... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4–1.** The State of a System Is Completely Specified by its Wave Function",
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In Chapter 3, we concluded that classical mechanical quantities are represented by linear operators in quantum mechanics. We now formalize this conclusion by our next postulate.
#### **Postulate 2**
*To every observable in classical mechanics there corresponds a linear operator in quantum mechanics.*
We have seen... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4–1.** The State of a System Is Completely Specified by its Wave Function",
"Header 3": "**4-2.** Quantum-Mechanical Operators Represent Classical-Mechanical Variables",
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The proof relies on the linear property of $\hat{A}$ (Section 3–2):
$$\begin{split} \hat{A}(c_1\phi_1 + c_2\phi_2) &= c_1\hat{A}\phi_1 + c_2\hat{A}\phi_2 \\ &= c_1a\phi_1 + c_2a\phi_2 = a(c_1\phi_1 + c_2\phi_2) \end{split}$$
#### EXAMPLE 4-3
Consider the eigenvalue problem
$$\frac{d^2\Phi(\phi)}{d\phi^2} = -m... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4–1.** The State of a System Is Completely Specified by its Wave Function",
"Header 3": "**4-2.** Quantum-Mechanical Operators Represent Classical-Mechanical Variables",
"token_count": 518,
"source_pdf": "datasets/webs... |
*In any measurement of the observable associated with the operator A, the only values that will ever be observed are the eigenvalues an' which satisfy the eigenvalue equation*
$$\hat{A}\psi_n = a_n \psi_n \tag{4.8}$$
Thus, in any experiment designed to measure the observable corresponding to *A,* the only values we... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4-3.** Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators",
"Header 3": "**Postulate 3**",
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SOLUTION: The operator that corresponds to the observable E is the Hamiltonian operator, which for a particle in a box is [Equation 3.14 with V(x) = 0]
$$\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$
The average energy is given by
$$\langle E \rangle = \int_0^a \psi_n^*(x) \hat{H} \psi_n(x) dx$$
$$= \frac{... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4-3.** Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators",
"Header 3": "**Postulate 3**",
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To this point, we have tacitly used all the given postulates in Chapter 3, and so our discussion so far should be fairly familiar. Now we must discuss the time dependence of wave functions. The time dependence of wave functions is governed by the timedependent Schrodinger equation. We cannot derive the time-dependent S... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4-4.** The Time Dependence of Wave Functions Is Governed by the Time-Dependent Schrodinger Equation",
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Table 4.1 contains a list of some commonly occurring quantum mechanical operators. We stated previously that these operators must have certain properties. We noticed they all are linear, and, in fact, linearity is a requirement we impose. A more subtle requirement arises if we consider Postulate 3, which says that, in ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "4-5. The Eigenfunctions of Quantum Mechanical Operators Are Orthogonal",
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The functions f(x) and g(x) are solutions to the one-dimensional harmonic oscillator problem discussed in detail in the next chapter.] Therefore,
$$\hat{A}g(x) = -i\hbar \frac{d}{dx} \frac{2^{1/2}}{\pi^{1/4}} x e^{-x^2/2}$$
$$= -i\hbar \frac{2^{1/2}}{\pi^{1/4}} [e^{-x^2/2} - x^2 e^{-x^2/2}]$$
and
$$\int_{\text{... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "4-5. The Eigenfunctions of Quantum Mechanical Operators Are Orthogonal",
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When two operators act sequentially on a function, f(x), such as in $\hat{A}\hat{B}f(x)$ , we apply each operator in turn, working from right to left (as in Example 3–5):
$$\hat{A}\,\hat{B}\,f(x) = \hat{A}[\,\hat{B}\,f(x)\,]$$
An important difference between operators and ordinary algebraic quantities is that oper... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4–6.** The Physical Quantities Corresponding to Operators That Commute Can Be Measured Simultaneously to Any Precision",
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Let's consider as an example, the simultaneous measurement of the momentum and position of a particle, so that *A* = *Px* and *B* = *X* in Equation 4.44. Equation 4.42 tells us that *[Px,* X] = *-ini,* and so Equation 4.44 gives
$$\sigma_{p}\sigma_{x} \geq \frac{1}{2} \left| \int \psi^{*}(x)(-i\hbar \hat{I})\psi(x)... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**4–6.** The Physical Quantities Corresponding to Operators That Commute Can Be Measured Simultaneously to Any Precision",
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4-1. Which of the following candidates for wave functions are normalizable over the indicated intervals?
**a.**
$$e^{-x^2/2}$$
$(-\infty, \infty)$ **b.** $e^x$ $(0, \infty)$ **c.** $e^{i\theta}$ $(0, 2\pi)$ **d.** $\sinh x$ $(0, \infty)$
**e.**
$$xe^{-x}$$
$(0, \infty)$
Normalize those that can be nor... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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$$\begin{array}{ccc}
\hat{A} & \hat{B} \\
\hline
\text{(a)} & \frac{d}{dx} & \frac{d^2}{dx^2} + 2\frac{d}{dx} \\
\text{(b)} & x & \frac{d}{dx} \\
\text{(c)} & SQR & SQRT \\
\text{(d)} & x^2\frac{d}{dx} & \frac{d^2}{dx^2}
\end{array}$$
**4-15.** In ordinary algebra, $(P+Q)(P-Q)=P^2-Q^2$ . Expand $(\hat{P}+\hat{Q})... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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Consider the two eigenvalue equations
$$\hat{A}\psi_n = a_n\psi_n$$
and $\hat{A}\psi_m = a_m\psi_m$
Multiply the first equation by 1/r~ and integrate; then take the complex conjugate of the second, multiply by *1jr n,* and integrate. Subtract the two resulting equations from each other to get
$$\int_{-\infty}^{\... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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Mathematically, we have
$$V(x) = V_0 \qquad 0 < x < a$$
$$0 \qquad x > a$$
Show that if $E < V_0$ , the solution to the Schrödinger equation in each region is given by
$$\psi_1(x) = Ae^{ik_1x} + Be^{-ik_1x} \qquad x < 0 \tag{1}$$
$$\psi_2(x) = Ce^{k_2x} + De^{-k_2x} \qquad 0 ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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A plot of the probability that a particle of energy E will penetrate a barrier of height $V_0$ plotted against the ratio $E/V_0$ (Equation 9 of Problem 4–35).
- **4-36.** Use the result of Problem 4–35 to determine the probability that an electron with a kinetic energy $8.0 \times 10^{-21}$ J will tunnel throug... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"Header 3": "FIGURE 4.3",
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Although Cartesian coordinates (x, *y,* and z) are suitable for many problems, there ?fe many other problems for which they prove to be cumbersome. A particularly i}nportant type of such a problem occurs when the system being described has some sort of a natural center, as in the case of an atom, where the (heavy) nucl... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**SPHERICAL COORDINATES**",
"token_count": 2044,
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Frequently the integrand in Equation D.7 will be a function only of r, in which case we say that the integrand is spherically symmetric. Let's look at Equation D.7 when $F(r, \theta, \phi) = f(r)$ :
$$I = \int_0^\infty dr r^2 \int_0^\pi d\theta \sin\theta \int_0^{2\pi} d\phi f(r)$$
(D.9)
Because f(r) is independ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**SPHERICAL COORDINATES**",
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- **D-1.** Derive Equations D.2 from D.1.
- **D-2.** Express the following points given in Cartesian coordinates in terms of spherical coordinates.
$$(x, y, z)$$
: $(1, 0, 0)$ ; $(0, 1, 0)$ ; $(0, 0, 1)$ ; $(0, 0, -1)$
**D-3.** Describe the graphs of the following equations:
**a.**
$$r = 5$$
,
**b.**
$$\the... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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Consider a mass *m* connected to a wall by a spring as shown in Figure 5.1. Suppose further that no gravitatio!].al force is acting on *m* so that the only force is due to
**FIGURE 5.1**
A mass connected to a wall by a spring. If the force acting upon the mass is directly proportiona... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-1.** A Harmonic Oscillator Obeys Hooke's Law",
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The simple harmonic oscillator is a good model for a vibrating diatomic molecule. A diatomic molecule, however, does not look like the system pictured in Figure 5.1 but more like two masses connected by a spring as in Figure 5.4. In this case we have two equations of motion, one for each mass:
$$m_1 \frac{d^2 x_1}{dt... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5–2.** The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule",
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Before we discuss the quantum-mechanical treatment of a harmonic oscillator, we should discuss how good an approximation it is for a vibrating diatomic molecule. The internuclear potential for a diatomic molecule is illustrated by the solid line in Figure 5.5. Notice that the curve rises steeply to the left of the mini... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-3.** The Harmonic-Oscillator Approximation Results from the Expansion of an Internuclear Potential Around its Minimum",
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This uniform spacing between energy levels is a property peculiar to the quadratic potential of a harmonic oscillator. Note also that the energy of the ground state, the state with $\nu=0$ , is $\frac{1}{2}\hbar\nu$ and is not zero as the lowest classical energy is. This energy is called the *zero-point energy* of t... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-3.** The Harmonic-Oscillator Approximation Results from the Expansion of an Internuclear Potential Around its Minimum",
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We will discuss molecular spectroscopy in some detail in Chapter 13, but here we will discuss the spectroscopic predictions of a harmonic oscillator. If we model the potential energy function of a diatomic molecule as a harmonic oscillator, then according to Equation 5.27, the vibrational energy levels of the diatomic ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5–5.** The Harmonic Oscillator Accounts for the Infrared Spectrum of a Diatomic Molecule",
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We will also see in Chapter 13 that not only must L'l v = ± 1 in the harmonic-oscillator model but the dipole moment of the molecule must change as the
**TABLE 5.1** The fundamental vibrational frequencies, the force constants, and bond lengths of some diatomic molecules
| M<br>l<br>l<br>o<br>e<br>c<br>u<br>e | 1<... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5–5.** The Harmonic Oscillator Accounts for the Infrared Spectrum of a Diatomic Molecule",
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I
The wave functions corresponding to the *E v* for a harmonic oscillator are nondegenerate and are given by
$$\psi_{v}(x) = N_{v} H_{v}(\alpha^{1/2} x) e^{-\alpha x^{2}/2}$$
(5.35)
where
$$\alpha = \left(\frac{k\mu}{\hbar^2}\right)^{1/2} \tag{5.36}$$
**TABLE 5.2** The first few Hermite polynomials.
| $H_0(... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-6.** The Harmonic-Oscillator Wave Functions Involve Hermite Polynomials",
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Recall from MathChapter B that an even function is a function that satisfies
$$f(x) = f(-x) \qquad \text{(even)} \tag{5.38}$$
and an odd function is one that satisfies
$$f(x) = -f(-x) \qquad \text{(odd)} \tag{5.39}$$
#### **EXAMPLE 5-6**
Show that the Hermite polynomials *H"* (~) are even if *v*is even and od... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-6.** The Harmonic-Oscillator Wave Functions Involve Hermite Polynomials",
"Header 3": "5–7. Hermite Polynomials Are Either Even or Odd Functions",
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"source_pdf": "datasets/websources/biochem/F814B... |
The final result is
$$\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)_{\theta, \phi} + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right)_{r, \phi} + \frac{1}{r^2 \sin^2 \theta} \left( \frac{\partia... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-6.** The Harmonic-Oscillator Wave Functions Involve Hermite Polynomials",
"Header 3": "5–7. Hermite Polynomials Are Either Even or Odd Functions",
"token_count": 1282,
"source_pdf": "datasets/websources/biochem/F814B... |
The allowed energies of a rigid rotator are given by Equation 5.57. We will prove in ChapteW,l.rhat electromagnetic radiation can cause a rigid rotator to undergo transitions from one state to another, and, in particular, we will prove that the selection rule for the rigid rotator says that transitions are allowed only... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**5-6.** The Harmonic-Oscillator Wave Functions Involve Hermite Polynomials",
"Header 3": "5-9. The Rigid Rotator Is a Model for a Rotating Diatomic Molecule",
"token_count": 1634,
"source_pdf": "datasets/websources/bioc... |
- **5-1.** Verify that $x(t) = A \sin \omega t + B \cos \omega t$ , where $\omega = (k/m)^{1/2}$ is a solution to Newton's equation for a harmonic oscillator.
- **5-2.** Verify that $x(t) = C \sin(\omega t + \phi)$ is a solution to Newton's equation for a harmonic oscillator.
- 5-3. The general solution for the ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1912,
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Calculate the fundamental vibrational frequency and the zero-point energy of <sup>79</sup>Br<sup>79</sup>Br.
- **5-15.** Verify that $\psi_1(x)$ and $\psi_2(x)$ given in Table 5.3 satisfy the Schrödinger equation for a harmonic oscillator.
- **5-16.** Show explicitly for a harmonic oscillator that $\psi_0(x)$ is ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1952,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Consider the transformation from Cartesian coordinates to plane polar coordinates where
$$x = r \cos \theta$$
$r = (x^2 + y^2)^{1/2}$
$y = r \sin \theta$ $\theta = \tan^{-1} \left(\frac{y}{x}\right)$ (1)
If a function $f(r, \theta)$ depends upon the polar coordinates r and $\th... | {
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"Header 2": "**Problems**",
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**5-37.** Compute the value of $\hat{L}^2Y(\theta,\phi)$ for the following functions:
**a.**
$$1/(4\pi)^{1/2}$$
**b.**
$$(3/4\pi)^{1/2}\cos\theta$$
**c.**
$$(3/8\pi)^{1/2} \sin \theta e^{i\phi}$$
**d.**
$$(3/8\pi)^{1/2} \sin \theta e^{-i\phi}$$
Do you find anything interesting about the results?
Problem... | {
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What is the value of the Rydberg constant for a deuterium atom?
- **5-47.** Calculate the ratio of the frequencies of the lines in the spectra of atomic deuterium and atomic hydrogen.
**Niels Bohr** was born in Copenhagen, Denmark, on October 7, 1885 and died there in 1962. In 1911, Bo... | {
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"Header 2": "**Problems**",
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For our model of a hydrogen atom, we will picture it as a proton fixed at the origin and an electron of mass me interacting with the proton through a Coulombic potential:
$$V(r) = -\frac{e^2}{4\pi\varepsilon_0 r} \tag{6.1}$$
where *e* is the charge on the proton, *s0* is the permittivity of free space, and *r* is t... | {
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"Header 2": "**6-1.** The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly",
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To solve Equation 6.10, we again use the method of separation of variables and let
$$Y(\theta, \phi) = \Theta(\theta)\Phi(\phi) \tag{6.11}$$
If we substitute Equation 6.11 into Equation 6.10 and divide by $\Theta(\theta)\Phi(\phi)$ , we find
$$\frac{\sin \theta}{\Theta(\theta)} \frac{d}{d\theta} \left( \sin \the... | {
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"Header 2": "**6–2.** The Wave Functions of a Rigid Rotator Are Called Spherical Harmonics",
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The factors in front of the $P_l(x)$ are chosen such that $P_l(1) = 1$ . In addition, although we will not prove it, it can be shown generally that the $P_l(x)$ in Table 6.1 are orthogonal, or that
$$\int_{-1}^{1} P_{l}(x) P_{n}(x) dx = 0 \qquad l \neq n$$
(6.24)
Keep in mind here that the limits on x correspo... | {
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According to Equation 6.31, the $Y_l^m(\theta, \phi)$ are orthonormal over the surface of a sphere and so are called *spherical harmonics*. The first few spherical harmonics are given in Table 6.3.
**TABLE 6.3** The first few spherical harmonics.
$$Y_{0}^{0} = \frac{1}{(4\pi)^{1/2}} \qquad Y_{1}^{0} = \left(\frac... | {
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SOLUTION: Using $Y_1^{-1}(\theta, \phi)$ from Table 6.3, the normalization condition is
$$\int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi \ Y_1^{-1}(\theta, \phi)^* Y_1^{-1}(\theta, \phi) = \frac{3}{8\pi} \int_0^{\pi} d\theta \sin \theta \sin^2 \theta \int_0^{2\pi} d\phi \stackrel{?}{=} 1$$
Letting $x = \... | {
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In this section, we will explore some of the quantum-mechanical properties of angular momentum. Recall that angular momentum is a vector quantity. The quantum-mechanical operators corresponding to the three components of angular momentum are given in Table 4.1. These operators are obtained from the classical expression... | {
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"Header 2": "**6–3.** Precise Values of the Three Components of Angular Momentum Cannot Be Measured Simultaneously",
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SOLUTION: Using $\hat{L}^2$ from Equation 6.32 and $\hat{L}_z$ from Equation 6.37, we have
$$\begin{split} \hat{L}^2 \hat{L}_z f &= -\hbar^2 \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial... | {
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"Header 2": "**6–3.** Precise Values of the Three Components of Angular Momentum Cannot Be Measured Simultaneously",
"token_count": 2043,
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*<sup>z</sup>*
S 0 L UTI 0 N : From the cone in Figure 6.1, we see that the *x-y* projection will be a circle. To determine the radius, *r,* of the circle, consider the *x, z* cross section
Because we have a right triangle, *r<sup>2</sup>*+ Tz *<sup>2</sup>*=*'2h <sup>2</sup>*and so *... | {
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Up to now we have solved Equation 6.9, giving the angular part of the hydrogen atomic orbitals. Now we will solve Equation 6.8, giving the radial part of the hydrogen atomic orbitals. Equation 6.8 with $\beta$ set equal to l(l+1) can be written as
$$-\frac{\hbar^2}{2m_a r^2} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \r... | {
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"Header 2": "**6–4.** Hydrogen Atomic Orbitals Depend upon Three Ouantum Numbers",
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$$n = 1, \qquad l = 0, \qquad m = 0 \qquad \psi_{100} = \frac{1}{\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2} e^{-\sigma}$$
$$n = 2, \qquad l = 0, \qquad m = 0 \qquad \psi_{200} = \frac{1}{\sqrt{32\pi}} \left(\frac{Z}{a_0}\right)^{3/2} (2 - \sigma) e^{-\sigma/2}$$
$$l = 1, \qquad m = 0 \qquad \psi_{210} = \frac{1... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**6–4.** Hydrogen Atomic Orbitals Depend upon Three Ouantum Numbers",
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Equation 6.49 tells us that the hydrogen atomic wave functions depend upon three quantum numbers, n, l, and m. The quantum number n is called the *principal quantum number* and has the values $1, 2, \ldots$ . The energy of the hydrogen atom depends upon only the principal quantum number through the equation $E_n = -e... | {
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"Header 2": "**6–4.** Hydrogen Atomic Orbitals Depend upon Three Ouantum Numbers",
"Header 3": "**6–5.** *s* Orbitals Are Spherically Symmetric",
"token_count": 1949,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf... |
For example, the normalization condition is
$$\int_{0}^{\infty} dr r^{2} \int_{0}^{\pi} d\theta \sin \theta \int_{0}^{2\pi} d\phi \, \psi_{1s}^{*}(r,\theta,\phi) \psi_{1s}(r,\theta,\phi) = 1$$
The hydrogen atomic wave functions are called *orbitals*, and, in particular, Equation 6.54 describes the 1s orbital; an el... | {
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"Header 2": "**6–4.** Hydrogen Atomic Orbitals Depend upon Three Ouantum Numbers",
"Header 3": "**6–5.** *s* Orbitals Are Spherically Symmetric",
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"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"... |
When $l \neq 0$ , the hydrogen atomic wave functions are not spherically symmetric; they depend on $\theta$ and $\phi$ . In this section, we will concentrate on the angular parts of the hydrogen wave functions. Let's first consider states with l = 1, or p orbitals. Because m = 0 or $\pm 1$ when l = 1, there are t... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**6–6.** There Are Three p Orbitals for Each Value of the Principal Quantum Number, $n \\ge 2$",
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The customary linear combinations are (Problem 6–42)
$$d_{z^{2}} = Y_{2}^{0} = \left(\frac{5}{16\pi}\right)^{1/2} (3\cos^{2}\theta - 1)$$
$$d_{xz} = \frac{1}{\sqrt{2}}(Y_{2}^{1} + Y_{2}^{-1}) = \left(\frac{15}{4\pi}\right)^{1/2} \sin\theta \cos\theta \cos\phi$$
$$d_{yz} = \frac{1}{\sqrt{2}i}(Y_{2}^{1} - Y_{2}^{-1... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**6–6.** There Are Three p Orbitals for Each Value of the Principal Quantum Number, $n \\ge 2$",
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The next system to study is clearly the helium atom, whose Schrödinger equation is
$$\left(-\frac{\hbar^{2}}{2M}\nabla^{2} - \frac{\hbar^{2}}{2m_{e}}\nabla_{1}^{2} - \frac{\hbar^{2}}{2m_{e}}\nabla_{2}^{2}\right)\psi(\mathbf{R}, \mathbf{r}_{1}, \mathbf{r}_{2}) + \left(-\frac{2e^{2}}{4\pi\varepsilon_{0}|\mathbf{R} - \m... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**6–7.** The Schrödinger Equation for the Helium Atom Cannot Be Solved Exactly",
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- **6-1.** Show that both $\hbar^2 \nabla^2 / 2m_a$ and $e^2 / 4\pi \varepsilon_0 r$ have the units of energy (joules).
- **6-2.** In terms of the variable $\theta$ , Legendre's equation is
$$\sin\theta \frac{d}{d\theta} \left( \sin\theta \frac{d\Theta(\theta)}{d\theta} \right) + (\beta^2 \sin^2\theta - m^2)\The... | {
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"Header 2": "**Problems**",
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Show that, in general
$$\frac{d^n I_0}{d\beta^n} = (-1)^n \int_0^\infty r^n e^{-\beta r} dr = (-1)^n I_n$$
$$= (-1)^n \frac{n!}{\beta^{n+1}}$$
and that
$$I_n = \frac{n!}{\beta^{n+1}}$$
- **6-23.** Prove that the average value of r in the 1s and 2s states for a hydrogenlike atom is $3a_0/2Z$ and $6a_0/Z$ , re... | {
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"Header 2": "**Problems**",
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Problem 5–23).
- **6-39.** The average value of r for a hydrogenlike atom can be evaluated in general and is given by
$$\left\langle r\right\rangle _{nl}=\frac{n^{2}a_{_{0}}}{Z}\left\{ 1+\frac{1}{2}\left[ 1-\frac{l(l+1)}{n^{2}}\right]\right\}$$
Verify this formula explicitly for the $\psi_{211}$ orbital.
**6-40... | {
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For light whose electric field vector is perpendicular to the direction of the external magnetic field, the selection rule is i".m = <sup>±</sup>1. In each case, how many of the possible transitions are allowed?
*Problems 6-48 through 6-57 develop the quantum-mechanical properties of angular momentum using operator n... | {
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Let *amax* be the largest possible value of *a± m1i.* By definition then, we have that
$$L_z \psi_{\alpha_{\max}\beta} = \alpha_{\max} \psi_{\alpha_{\max}\beta}$$
$$\hat{L}^2 \psi_{\alpha_{\max}\beta} = \beta^2 \psi_{\alpha_{\max}\beta}$$
and
$$\hat{L}_+ \psi_{\alpha_{--}\beta} = 0$$
Operate on the last equat... | {
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In Chapter 7, we will encounter *n* linear algebraic equations *inn* unknowns. Such equations are best solved by means of determinants, which we discuss **in** this MathChapter. Consider the pair of linear algebraic equations
$$a_{11}x + a_{12}y = d_1$$
$a_{21}x + a_{22}y = d_2$ (E.1)
If we multiply the first of... | {
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"Header 2": "**DETERMINANTS**",
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SOLUTION: Expand about the first row of elements to obtain
$$\begin{vmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x \end{vmatrix} - \begin{vmatrix} 1 & 1 & 0 \\ 0 & x & 1 \\ 0 & 1 & x \end{vmatrix} = 0$$
Now expand about the first column of each of the $3 \times 3$ determinants to obtain
$$(x)(x)\begin{vmatrix} ... | {
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"Header 2": "**DETERMINANTS**",
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The determinant of the coefficients of x and y is
$$D = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}$$
According to Rule 4.
$$\begin{vmatrix} a_{11}x & a_{12} \\ a_{21}x & a_{22} \end{vmatrix} = xD$$
Furthermore, according to Rule 6,
$$\begin{vmatrix} a_{11}x + a_{12}y & a_{12} \\ a_{21}x ... | {
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"Header 2": "**DETERMINANTS**",
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**E-1** . Evaluate the determinant
$$D = \begin{vmatrix} 2 & 1 & 1 \\ -1 & 3 & 2 \\ 2 & 0 & 1 \end{vmatrix}$$
Add column 2 to column 1 to get
$$\begin{array}{c|ccccc}
3 & 1 & 1 \\
2 & 3 & 2 \\
2 & 0 & 1
\end{array}$$
and evaluate it. Compare your result with the value of D. Now add row 2 of D to row **1** of D... | {
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•
We will first illustrate the *variational method.* Consider the ground state of some arbitrary system. The ground-state wave function 1/1*0*and energy *E0* satisfy the SchrOdinger equation
$$\hat{H}\psi_0 = E_0\psi_0 \tag{7.1}$$
Multiply Equation 7.1 from the left by 1/1; and integrate over all space to obtain ... | {
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#### FIGURE 7.1
A comparison of the optimized Gaussian trial wave function $\phi(r)=(2\alpha/\pi)^{3/4}e^{-\alpha r^2}$ , where $\alpha$ is given by Equation 7.8 (dashed line), and the exact ground-state hydrogen wave function, $\psi(r)=(1/\pi a_0^3)^{1/2}e^{-r/a_0}$ , where $a_0... | {
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"Header 2": "7-1. The Variational Method Provides an Upper Bound to the Ground-State Energy of a System",
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Both functions are plotted against $x/(\hbar^2/\mu k)^{1/4}$ , and the vertical axis is expressed in units of $(\alpha/\pi)^{1/4}$ , where $\alpha = (k\mu)^{1/2}/\hbar$ (see Example 7–3).
In Figure 7.2, the normalized, optimized trial function of Example 7–2 is compared with the exact ground-state harmonic oscill... | {
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As another example of the variational method, consider a particle in a one-dimensional box. Even without prior knowledge of the exact ground-state wave function, we should expect it to be symmetric about x = a/2 and to go to zero at the walls. One of the simplest functions with these properties is $x^n(a-x)^n$ , where... | {
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"Header 2": "**7–2.** A Trial Function That Depends Linearly on the Variational Parameters Leads to a Secular Determinant",
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In this case,
$$f_1 = x(1-x)$$
and $f_2 = x^2(1-x)^2$ (7.38)
and the matrix elements (see Equations 7.27 and 7.31) are (see Problem 7–26)
$$H_{11} = \frac{\hbar^2}{6m}$$
$$S_{11} = \frac{1}{30}$$
$$H_{12} = H_{21} = \frac{\hbar^2}{30m}$$
$$S_{12} = S_{21} = \frac{1}{140}$$
$$H_{22} = \frac{\hbar^2}{105m... | {
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If we use a linear combination of N functions as in Equation 7.25 instead of using a linear combination of two functions as we have done so far, then we obtain N simultaneous linear algebraic equations for the $c_i$ s:
$$c_{1}(H_{11} - ES_{11}) + c_{2}(H_{12} - ES_{21}) + \dots + c_{N}(H_{1N} - ES_{1N}) = 0$$
$$... | {
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It is a fairly common practice to use a trial function of the form
$$\phi = \sum_{j=1}^{N} c_j f_j$$
where the $f_j$ themselves contain variational parameters. An example of such a trial function for the hydrogen atom is
$$\phi = \sum_{j=1}^{N} c_j e^{-\alpha_j r^2}$$
where the $c_j$ s and the $\alpha_j$ s ... | {
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"Header 2": "**7–3.** Trial Functions Can Be Linear Combinations of Functions that Also Contain Variational Parameters",
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The idea behind perturbation theory is the following. Suppose that we are unable to solve the Schrödinger equation
$$\hat{H}\psi = E\psi \tag{7.41}$$
for some system of interest but that we do know how to solve it for another system that is in some sense similar. We can write the Hamiltonian operator in Equation 7.... | {
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"Header 2": "**7–4.** Perturbation Theory Expresses the Solution to One Problem in Terms of Another Problem Solved Previously",
"token_count": 1959,
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The energy levels are given through first order by
$$E = \frac{n^2 h^2}{8ma^2} + \frac{V_0}{2} + O(V_0^2) \qquad n = 1, 2, 3, \dots$$
where the term $O(V_0^2)$ emphasizes that terms of order $V_0^2$ and higher have been dropped. Thus, we see in this case that each of the unperturbed energy levels is shifted by ... | {
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**7-1.** This problem involves the proof of the variational principle, Equation 7.4. Let $\hat{H}\psi_n = E_n\psi_n$ be the problem of interest, and let $\phi$ be our approximation to $\psi_0$ . Even though we do not know the $\psi_n$ , we can express $\phi$ formally as
$$\phi = \sum_{n} c_n \psi_n \tag{1}$$ ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
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The Hamiltonian operator for this system is (see Equation 6.43)
$$\hat{H} = -\frac{\hbar^2}{2mr^2} \frac{d}{dr} \left( r^2 \frac{d}{dr} \right) + \frac{\hbar^2 l(l+1)}{2mr^2} \qquad 0 < r \le a$$
In the ground state, l = 0 and so
$$\hat{H} = -\frac{\hbar^2}{2mr^2} \frac{d}{dr} \left( r^2 \frac{d}{dr} \right) \qqu... | {
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The general result is that terms in a trial function that correspond to higher and higher energies contribute less and less to the total ground-state energy.
**7-19.** We will derive the equations for first-order perturbation theory in this problem. The problem we want to solve is
$$\hat{H}\psi = E\psi \tag{1}$$
... | {
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(*Hint*: Use the expansion $e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots$ .) What is the Hamiltonian operator for the Morse potential? Show that the Hamiltonian operator can be written in the form
$$\hat{H} = -\frac{\hbar^2}{2\mu} \frac{d^2}{dx^2} + ax^2 + bx^3 + cx^4 + \cdots$$
(1)
How are the constants a, b, and c... | {
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Using the law of cosines,
$$r_{12} = (r_1^2 + r_2^2 - 2r_1r_2\cos\theta)^{1/2}$$
show that £<ll becomes
$$E^{(1)} = \frac{e^2}{4\pi\varepsilon_0} \frac{Z^6}{a_0^6 \pi^2} \int_0^\infty dr_1 e^{-2Zr_1/a_0} 4\pi r_1^2 \int_0^\infty dr_2 e^{-2Zr_2/a_0} r_2^2$$
$$\times \int_0^{2\pi} d\phi \int_0^\pi \frac{d\theta \... | {
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"Header 2": "**Problems**",
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"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Show that
$$E^{(1)} = \frac{e^2}{4\pi\varepsilon_0} \frac{16Z^6}{a_0^6} \int_0^\infty dr_1 r_1^2 e^{-2Zr_1/a_0} \int_0^\infty dr_2 r_2^2 \frac{e^{-2Zr_2/a_0}}{r_2}$$
Now show that
$$\begin{split} E^{(1)} &= \frac{e^2}{4\pi\varepsilon_0} \frac{16Z^6}{a_0^6} \int_0^\infty dr_1 r_1 e^{-2Zr_1/a_0} \int_0^{r_1} dr_2 r... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 2032,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
We concluded Chapter 6 with an introduction to the helium atom. We showed there that if.we considered the nucleus to be fixed at the origin, then the Schrodinger equation has the form
$$\left\{\hat{H}_{H}(1) + \hat{H}_{H}(2) + \frac{e^{2}}{4\pi\varepsilon_{0}r_{12}}\right\}\psi(\mathbf{r}_{1}, \mathbf{r}_{2}) = E\psi... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Multielectron Atoms**",
"token_count": 338,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
We will apply both perturbation theory and the variational method to the helium atom, but before doing so, we will introduce a system of units, called atomic units, that is widely used in atomic and molecular calculations to simplify the equations. Natural units of mass and charge on an atomic or molecular scale are th... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Multielectron Atoms**",
"Header 3": "**8-1.** Atomic and Molecular Calculations Are Expressed in Atomic Units",
"token_count": 1923,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The problem we want to solve is $\hat{H}\psi = E\psi$ , where $\hat{H}$ is given by Equation 8.5. We applied perturbation theory to this problem at the end of Section 7–4 by considering the
**TABLE 8.2** Ground-state energy of the helium atom<sup>a</sup>.
| Method ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–2.** Both Perturbation Theory and the Variational Method Can Yield Excellent Results for Helium",
"token_count": 2020,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
#### **EXAMPLE 8-2**
Show that $S_{nlm}(r, \theta, \phi)$ is not orthogonal to $S_{n'lm}(r, \theta, \phi)$ .
SOLUTION: We must show that $I \neq 0$ where
$$I = \int_0^\infty dr r^2 \int_0^\pi d\theta \sin\theta \int_0^{2\pi} d\phi \, S_{nlm}^*(r,\theta,\phi) S_{n'lm}(r,\theta,\phi)$$
$$= \int_0^\infty dr r... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–2.** Both Perturbation Theory and the Variational Method Can Yield Excellent Results for Helium",
"token_count": 1325,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The starting point of the Hartree-Fock procedure for helium is to write the two-electron wave function as a product of orbitals, as in Equation 8.15:
$$\psi(\mathbf{r}_1, \mathbf{r}_2) = \phi(\mathbf{r}_1)\phi(\mathbf{r}_2) \tag{8.17}$$
The two functions on the right side of Equation 8.17 are the same because we ar... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8-3.** Hartree-Fock Equations Are Solved by the Self-Consistent **Field** Method",
"token_count": 1571,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Although the Schödinger equation was amazingly successful in predicting or explaining the results of most experiments, it could not explain a few phenomena. One was the doublet yellow line in the atomic spectrum of sodium. The Schrödinger equation predicts that there should be one line around 590 nm, whereas two closel... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8-3.** Hartree-Fock Equations Are Solved by the Self-Consistent **Field** Method",
"Header 3": "8-4. An Electron Has An Intrinsic Spin Angular Momentum",
"token_count": 1225,
"source_pdf": "datasets/websources/biochem/... |
We must now include the spin function with the spatial wave function. We postulate that the spatial and spin parts of a wave function are independent and so write
$$\Psi(x, y, z, \sigma) = \psi(x, y, z)\alpha(\sigma)$$
or $\psi(x, y, z)\beta(\sigma)$ (8.28)
The complete one-electron wave function $\Psi$ is call... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–5.** Wave Functions Must Be Antisymmetric in the Interchange of Any Two Electrons",
"token_count": 2019,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Another is
$$\begin{split} \int \int \alpha^*(\sigma_1) \beta^*(\sigma_2) \alpha(\sigma_2) \beta(\sigma_1) d\sigma_1 d\sigma_2 \\ &= \int \alpha^*(\sigma_1) \beta(\sigma_1) d\sigma_1 \int \beta^*(\sigma_2) \alpha(\sigma_2) d\sigma_2 = 0 \end{split}$$
The other two are equal to 1 and 0, and so
$$I = c^2 \int \Psi_... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–5.** Wave Functions Must Be Antisymmetric in the Interchange of Any Two Electrons",
"token_count": 212,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Now that we have introduced spin and have seen that we must use antisymmetric wave functions, we must ask why we could ignore the spin part of the wave function when we treated the helium atom in Sections 7–1 and 8–2. The reason is that $\Psi_2$ can be factored into a spatial part and a spin part, as we saw in Equati... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–6.** Antisymmetric Wave Functions Can Be Represented by Slater Determinants",
"token_count": 1928,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In Section 8–3, we discussed the Hartree-Fock method for the helium atom. The Hartree-Fock equation for this system is given by Equation 8.20, where $\hat{H}_1^{\text{eff}}$ is given by Equation 8.19. The helium atom is a special case because the Slater determinant factors into a spatial part and a spin part, and so ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–7.** Hartree-Fock Calculations Give Good Agreement with Experimental Data",
"token_count": 2008,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Electron configurations of atoms are ambiguous in the sense that a number of sets of $m_l$ and $m_s$ are consistent with a given electron configuration. For example, consider the ground-state electron configuration of a carbon atom, $1s^22s^22p^2$ . The two 2p electrons could be in any of the three 2p orbitals $(... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"token_count": 2010,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Similarly, because *ms* <sup>1</sup>and *ms*2can both have values of± 112, *Ms* can be -1, 0, or 1. We now set up a table with its columns headed by the possible values of *M5* and its rows headed by the possible values of *ML,* and we then fill in the sets of values of *<sup>m</sup>*<sup>11</sup> , *msl' m*<sup>12</su... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"token_count": 1657,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Using this information, we set up a table with its columns headed by the possible values of M s and its rows headed by the possible values of *M* L, and then fill in the microstates consistent with each value of *M* L and *M* s• as shown:
| | M<br>s | ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"token_count": 1965,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
| Electron configuration | Term symbol (excluding the <i>J</i> subscript) |
|------------------------|-------------------------------------------------------------------------------------------------------------------------|
| $s^1$ ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"token_count": 1684,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Each of the states designated by a term symbol corresponds to a determinantal wave function that is an eigenfunction of *i* 2 and S2 ' and each state corresponds to a certain energy. Although we could calculate the energy associated with each state, in practice, the various states are ordered according to three empiric... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"Header 3": "**8-10.** Hund's Rules Are Used to Determine the Term Symbol of the Ground Electronic State",
"token_count": 532,
"source_pd... |
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