Questions and Answers of An Introduction To Accounting 1st

Here is a list of several atoms and their ground states. For each case, determine \(J, L, S\) and show that the vector model is satisfied.(a) beryllium (Be): \({ }^{1} S_{0}\)(b) aluminum (Al): \({
Beginning in the 1920s, Russian physicist Pyotr Kapitza or Kapitsa (18941984, Nobel laureate in physics 1978) measured the Paschen-Back effect to an accuracy of 1 percent to 3 percent in various
Consider transitions from a \({ }^{2} D\) state to a \(2 P\) state in the strong field PaschenBack regime. List all allowed transitions and show that there are only three different spectral lines.
What is the longest wavelength of the Paschen series spectrum? Would it be visible to the human eye?
For the Lyman series, what is the wavelength of the line with the second longest wavelength? Would it be visible to the human eye?
In 1924 French theorist Louis de Broglie (1892 1987, Nobel laureate in physics 1929) proposed that matter has a wavelike nature expressed by \(\lambda=\frac{h}{m v}\). Show that his relation agrees
This problem presents an example of the Correspondence Principle. Suppose that the electron in hydrogen is traveling in a circular orbit with quantum number \(n\) and makes a transition to \(n+1\),
Write the Hamiltonian for the electron in a hydrogenic atom. Assume that the nucleus is fixed in position.
Write the Hamiltonian for the two electrons (labeled 1 and 2) in a helium atom. Assume that the nucleus is fixed in position.
A function \(y(t)\) is expressed as a Fourier series.Show that if \(y(t)\) is real, \(\phi_{-n}=\phi_{n}^{*}\). (1) = = 8118 +..
Use the quantization condition Eq. (5.13) to show that \(\mathbf{p q} \mathbf{q}^{2}-\mathbf{q}^{2} \mathbf{p}=-2 i \hbar \mathbf{q}\).Equation 5.13 h pq-qp= 1 (5.13) 2ri =-ihl,
Consider a particle moving freely in the absence of applied forces. Show that according to the principles of matrix mechanics its linear momentum is conserved.
Consider the electron in a hydrogen atom traveling in a circular orbit. Show that according to the principles of matrix mechanics the electron's angular momentum commutes with the Hamiltonian.What is
Consider the Hamiltonian for a 1-dimensional harmonic oscillator. According to the quantization condition, Eq. (5.13), pq - qp is a diagonal matrix. Show that this is a necessary condition by proving
A particle of mass \(m\) is moving freely along the \(x\)-axis (no external forces). What is its wave function?
Consider a 1-dimensional harmonic oscillator with mass \(m\) moving along the \(x\)-axis. Its angular frequency is \(\omega=\sqrt{\frac{k}{m}}\), where \(k\) is the spring constant. The wave function
Consider a 1-dimensional harmonic oscillator with mass \(m\) moving along the \(x\)-axis. Its angular frequency is \(\omega=\sqrt{\frac{k}{m}}\), where \(k\) is the spring constant. The wave function
The wave function for the ground state of hydrogen \(n=1\) is\[\psi(r)=\frac{1}{\sqrt{a_{0}^{3} \pi}} e^{-\frac{r}{a_{0}}}\]Take the reduced mass to be \(\approx m_{e}\).(a) Show that this wave
The wave function for the first excited state of hydrogen \(n=2, \ell=0\) is\[\psi(r)=\frac{1}{4 \sqrt{2 a_{0}^{3} \pi}}\left(2-\frac{r}{a_{0}}ight) e^{-\frac{r}{2 a_{0}}}\]Take the reduced mass to
The wave function for the first excited state of hydrogen \(n=2, \ell=0\) is\[\psi(r)=\frac{1}{4 \sqrt{2 a_{0}^{3} \pi}}\left(2-\frac{r}{a_{0}}ight) e^{-\frac{r}{2 a_{0}}}\]Take the reduced mass to
Consider the row vector \(\langle A|\) and column vector \(|Bangle\) :(a) Calculate the inner product \(\langle A \mid Bangle\).(b) Calculate the outer product \(|Bangle\langle A|\). 3 (A = (31 4
Consider the row vector \(\langle A|\) and column vector \(|Bangle\) :(a) Calculate the inner product \(\langle A \mid Bangle\).(b) Calculate the outer product \(|Bangle\langle A|\). 5 (A = (2 21-31)
For the rigid rotor dumbbell (arbitrary amplitude \(A\) ), normalize the wave function.
The wave function for one of the excited states of a freely rotating rigid diatomic molecule is\[=\sqrt{\frac{3}{8 \pi}} e^{i \phi} \sin \theta\]Use Schrödinger's wave equation to calculate the
Diagonalize the reducible matrix for the operation \(\mathbf{T} x=-x\) introduced.What are the irreducible representations? 0 (f )) 1
For the diagonalization example solve for the elements of matrix \(\mathbf{S}\) and normalize to obtain definite values. Several methods have been used to normalize a matrix. For this problem,
Consider the matrix.(a) Diagonalize the matrix.(b) Show that diagonalization does not change the character (trace) of the matrix. 2 (6+) 1
Consider two matrices \(S\) and \(T\).Evaluate \(S \otimes T\) and show that the result can be written in the form (a b S=( ) - ( b) A T B D
Consider the wave function.\[\left|\Psi_{A B}^{(e n t)}ightangle=\frac{1}{\sqrt{2}}\left(|\downarrowangle_{A} \otimes\left|\uparrow{ }_{B}ightangle-|\uparrowangle_{A}
Consider two matrices \(A\) and \(B\).Evaluate \(A \otimes B\) and \(B \otimes A\). Are they equal? A = (27) 4 B= 3
Consider the vector \(\mathbf{A}\).\[\mathbf{A}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\]Calculate\[\mathbf{A}^{\prime}=R\left(60^{\circ}, zight) \mathbf{A} \text {. }\]
Consider the vector \(\mathbf{A}\).\[\mathbf{A}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\]Calculate\[\mathbf{A}^{\prime}=R\left(-30^{\circ}, zight) \mathbf{A} .\]
Calculate the matrix for \(R(\theta, x)\) for clockwise rotation of the axes and also for counterclockwise rotation of a vector.
Calculate the matrix for \(R(\theta, y)\) for clockwise rotation of the axes and also for counterclockwise rotation of a vector.
The line \(y=x\) makes an angle of \(45^{\circ}\) with both the \(x\) - and \(y\)-axes. It is evident that rotating the line by \(\mathrm{R}\left(45^{\circ}, zight)\) makes it coincide with the
Consider the hyperbola.\[\frac{x^{2}}{4}-y^{2}=1\]Calculate its equation after rotation by \(\mathrm{R}\left(90^{\circ}, zight)\).
Give an argument to show that the three Pauli spin matrices do not form a group.
(a) Show that the three Pauli spin matrices are Hermitian.(b) Show that the square of each Pauli matrix is the identity.
For the Pauli matrices, show that\[\sigma_{i} \sigma_{j}=i P \sigma_{k}\]where the subscripts \(i, j, k\) are all different \(P=+1\) if the subscripts are an even permutation and -1 if the
Consider the spin vector \(\sigma_{S}\).\[\sigma_{S}=\frac{\hbar}{2}\left(\sigma_{x} \hat{\mathbf{i}}+\sigma_{y} \hat{\mathbf{j}}+\sigma_{z} \hat{\mathbf{k}}ight)\]Calculate the matrix
With reference to Eq. (7.12), show by calculation that the similarity transformation of matrix \(V\) by \(U_{y}(\beta)\) generates the rotation matrix \(R(\beta, y)\).Equation 7.12 OS (2) - sin Uy
Diagonalize this Pauli spin matrix,\[\frac{\hbar}{2}\left(\begin{array}{cc}0 & -i \\i & 0\end{array}ight)\]and find its eigenvalues.
Consider two operators \(\mathbf{A}\) and B. Show that\[e^{-i \mathbf{A}} e^{-i \mathbf{B}}=e^{-i(\mathbf{A}+\mathbf{B})}\]only if \(\mathbf{A}\) and \(\mathbf{B}\) commute.
Show that the angular momentum operators \(\left(I_{x}, I_{y}, I_{z}ight)\) are Hermitian.
Consider an operator \(\mathbf{U}=e^{i \theta \mathbf{G}}\) with generator \(\mathbf{G}\). If \(\mathbf{U}\) commutes with a Hamiltonian \(\mathbf{H}\), use the first-order method to show that
Show that the ladder operators \(I_{ \pm}\)are not Hermitian.
Show that \(\left[I^{2}, I_{ \pm}ight]=0\).
Find the eigenvalues of \(I_{x}\).\[I_{x}=\left(\begin{array}{ccc}0 & 0 & 0 \\0 & 0 & -i \\0 & i & 0\end{array}ight)\]
Suppose that an irreducible representation of the rotation group is represented by \(7 \times 7\) matrices.(a) What is the value of its angular momentum quantum number \(j\) ?(b) What are the
Suppose that an irreducible representation of the rotation group is represented by \(4 \times 4\) matrices.(a) What is the value of its angular momentum quantum number \(j\) ?(b) What are the
Evaluate \(I_{x}|j mangle\).
Evaluate \(I_{y}|j mangle\).
For \(j=1\), consider the basis functions \(u_{1 m}\). Show by calculation that \(I_{+} u_{10}\) is proportional to the spherical harmonic \(Y_{1}^{1}\).
Consider the basis function \(u_{11}\) for \(D^{(1)}(\alpha, \beta, \gamma)\). Calculate the result of rotating \(u_{11}\) through the Euler angles \(\gamma=0, \beta=\frac{\pi}{4},
Consider \(I^{2}=I_{x}^{2}+I_{y}^{2}+I_{z}^{2}\). Use Eqs. (8.27), (8.28), and (8.29) to show that \(I^{2} u_{j m}=j(j+1) u_{j m}\).Equations == I+|jm)=j(j + 1) m(m + 1)|j m + 1) I_ |j m) = j(j +
For \(\ell=2\), develop by calculation an expression proportional to \(Y_{2}^{-1}(\theta, \phi)\).
For \(\ell=3\), develop by calculation an expression proportional to \(Y_{3}^{-2}(x, y)\).
a) Evaluate \(e^{-i \phi I_{z}}|j mangle\).b) Use the result ofa) to evaluate the matrix the element \(\left\langle j m^{\prime}\left|e^{-i \phi I_{z}}ight| j mightangle\).
In the HD molecule of hydrogen, one nucleus is \({ }^{1} \mathrm{H}\left(\operatorname{spin} \frac{1}{2}ight)\) and the other is a deuteron \({ }^{2} \mathrm{H}\) (spin 1). What are the possible spin
In a molecule of "heavy hydrogen," \(\mathrm{D}_{2}\), the nuclei are both deuterons (spin 1). The two electrons are paired and assumed to be in a state \(j=0\). Referring to Table F.3 in Appendix F,
In the HD molecule of hydrogen, one nucleus is \({ }^{1} \mathrm{H}\left(\operatorname{spin} \frac{1}{2}ight)\) and the other is a deuteron \({ }^{2} \mathrm{H}\) (spin 1). The two electrons are
In a molecule of "heavy hydrogen," \(\mathrm{D}_{2}\), the nuclei are both deuterons (spin 1). The two electrons are paired and assumed to be in a state \(j=0\). Referring to Table F. 3 in Appendix
What angular momentum values are possible from the combination of \(j=2\) and \(j=1\) ? Draw sketches according to the vector model to illustrate your answers.
Use values from Table F.2 in Appendix F to illustrate the orthogonality relation Eq. (8.43). ()) 11/2 ()(1) Table F.2 3-j coefficients for D (1) D) ww W 3/2 1/2 1 1/2 (3) W -1/2 -1/2 W v) -3/2 -1/2
What angular momentum values are possible from the combination of \(j=\frac{3}{2}\) and \(j=\frac{1}{2}\) ? Draw sketches according to the vector model to illustrate your answers.
Use values from Table F.3 in Appendix F to illustrate the orthogonality relation Eq. (8.44).Data from Table 1-12 2 2 - In (1),,(1) (1),,(1) u(D) 6(1) (1),(1) a On (1) vj oa on In (1),,(1) uv (1)
Consider a state \(\psi_{\ell m}\) with \(\ell=2\) and \(m=1\), and a state with \(\ell=1\) and \(m=0\). Show, using spherical harmonics, whether an electric dipole transition is allowed between
Consider an atom with a single electron outside closed shells. Take the \(z\) component of a magnetic dipole transition with matrix element\[\left\langle\ell^{\prime}, m^{\prime},
An atom's electron distribution will have an electric quadrupole moment if the distribution is a nonspherical ellipsoid. The matrix element for an electric quadrupole transition is proportional to
The quadrupole operator \(Q\) discussed in Problem 27 is a second rank tensor \(\mathbf{T}_{q}^{(k)}\) with \(k=2\). Based on this, what are the selection rules \(\Delta j\) and \(\Delta m_{j}\) for
(a) Show that according to the classical theory of the Zeeman effect a possible motion is circular motion, either \(\mathrm{ccw}\) or \(\mathrm{cw}\), in the \(x-y\) plane perpendicular to the
(a) From Eq. (9.15) the Landé \(g\)-factor is\[\begin{equation*}g_{J}=1+\frac{J(J+1)-L(L+1)+S(S+1)}{2 J(J+1)} . \tag{9.15}\end{equation*}\]Show that \(g_{J}\) can be written in the equivalent
Show that the Landé \(g\) factor \(=1\) for \(S=0\).
Here is a list of several atoms and their ground states. For each case, determine \(J, L, S\) and show that the vector model is satisfied.(a) neon (Ne): \({ }^{1} S_{0}\)(b) silver ( \(\mathrm{Ag}):{
(a) The ground state of iron (Fe) is \({ }^{5} D_{4}\). What is the Landé \(g\)-factor of this state?(b) One of the excited states of \(\mathrm{Fe}\) is \({ }^{3} F_{2}\). What is the Lande
An excited state of mercury \((\mathrm{Hg})\) is \({ }^{3} P_{J}\), where the possible values of \(J\) are \(2,1,0\).(a) What is the Landé \(g\)-factor of the state \({ }^{3} P_{2}\) ?(b) What is
Consider transitions from a \({ }^{3} S_{1}\) state to a lower \({ }^{3} P_{1}\) state. In a magnetic field of \(3.2 \mathrm{~T}\) the transitions show weak-field Zeeman splitting. Calculate the span
The visible spectrum of sodium is dominated by two closely spaced bright yellow lines at \(589.0 \mathrm{~nm}\) and \(589.6 \mathrm{~nm}\). The line at \(589.0 \mathrm{~nm}\) is due to a transition
Taking into account only fine structure due to \(L S\) coupling, calculate the energy difference between a level characterized by \(J\) and the neighboring level \(J-1\). The result is known as the
Consider hyperfine splitting in the absence of a magnetic field. Calculate the energy difference between a level characterized by \(F\) and the neighboring level \(F-1\). The result is known as the
The ground state of sodium \({ }^{2} S_{1 / 2}\) is subject to hyperfine splitting into two levels with a separation of \(1.771 \times 10^{9} \mathrm{~Hz}\).(a) The atomic structure factor \(\alpha\)
Hyperfine splitting in \({ }^{1} \mathrm{H}\) is \(1420 \times 10^{6} \mathrm{~Hz}\) in the absence of a magnetic field where \(I . J . F . M_{F}\) are all good quantum numbers. In a moderately
Consider the spin states of three electrons \(\uparrow \uparrow \downarrow \downarrow \downarrow \uparrow \downarrow \uparrow \uparrow\). Treating these as basis functions for \(\mathbf{S}_{3}\),
Consider the spin states of three electrons \(\uparrow \uparrow \downarrow \downarrow \downarrow \uparrow \downarrow \uparrow \uparrow\). Treating these as basis functions for \(\mathbf{S}_{3}\),
Show that the Slater determinant wave function for two electrons is antisymmetric.
The ground state of \({ }_{8} \mathrm{O}\) has two \(1 s\) electrons, two \(2 s\), and four \(2 p\).(a) What is the complete electron configuration in standard notation?(b) What are the possible
The ground state of \({ }_{13} \mathrm{Al}\) has two \(3 s\) and one \(3 p\) electron outside closed shells.(a) What is the complete electron configuration in standard notation?(b) What is a possible
Radium-226 decays with a half-life of 1,620 years to radon-222 plus a helium nucleus ( \(\alpha\) particle) according to the nuclear reaction equation\[{ }_{88}^{226} \mathrm{Ra} ightarrow{
Consider an \(\alpha\) particle, which is the nucleus of \({ }_{2}^{4} \mathrm{He}\) atomic mass \(4.002603 \mathrm{u}\).(a) Use the data in Table 10.1 to calculate the binding energy per nucleon.(b)
Express the natural unit of length in terms of the SI unit meter.
Express the natural unit of energy in terms of the SI unit joule and also in terms of \(\mathrm{eV}\).
Express the natural unit of electric charge \(e\) in terms of the SI unit coulomb. Use the dimensionless fine structure constant \(=\frac{e^{2}}{4 \pi \epsilon_{0} \hbar c}\).
Express time in natural units (combined with \(\mathrm{GeV}\) ) in terms of the SI unit second.
A consequence of Yukawa's theory was a potential energy (in natural units),\[\frac{e^{-m r}}{r}\]where \(m\) is the mass of the mediating boson and \(r \approx 1.5 \mathrm{fm}\) is a measure of the
Making reasonable assumptions, use isospin to estimate the ratio of the total cross sections for the processes\[\begin{aligned}& \pi^{-}+p^{+} ightarrow \pi^{-}+p^{+} \\& \pi^{-}+p^{+}
Making reasonable assumptions, use isospin to estimate the ratio of the total cross sections for observable pion production processes\[\begin{gathered}p^{+}+p^{+} ightarrow \pi^{+}+d^{+} \^{0}+p^{+}
Suppose a mass \(m\) is thrown upward with initial velocity \(v_{0}\) at an angle \(\theta\) in a constant gravitational field.(a) Write the Lagrangian for this situation.(b) Use your Lagrangian to
Making reasonable assumptions, use isospin to estimate the ratio of the total cross sections for observable interactions of pions and deuteronswhere \(d\) is a deuteron. ++d+p+p+
In special relativity, the relation between energy EE, momentum pp, and rest mass m0m0 is\[E^{2}=p^{2}+m_{0}^{2}\]written in natural units c=1c=1.(a) Insert factors of cc as necessary to write this
The neutron is made of \(u\) and \(d\). What is its composition according to the quark model?

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