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physics
modern physics

Questions and Answers of Modern Physics

Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that istrue.
A hydrogen atom in a state having a binding energy (the energy required to remove an electron) of 0.85 eV makes a transition to a state with an excitation energy (the difference between the energy of
The wave functions for the three states with the dot plots shown in figure, which have n = 2, ? = 1, and m? = 0, +1, and ? 1, are in which the subscripts on ?(r, ?) give the values of the quantum
Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a, where a is the Bohr radius.
What is the probability that an electron in the ground state of the hydrogen atom will be found between two spherical shells whose radii are r and r + ∆r, (a) if r = 0.500a and ∆r =
Light of wavelength 102.6 nm is emitted by a hydrogen atom. What are the (a) Higher quantum number and (b) Lower quantum number of the transition producing this emission?(c) What is the name of the
For what value of the principal quantum number n would the effective radius, as shown in a probability density dot plot for the hydrogen atom, be 1.0 mm? Assume that ( has its maximum value of n = 1.
The wave function for the hydrogen-atom quantum state represented by the dot plot shown in figure, which has n = 2 and ? = m? = 0, is in which a is the Bohr radius and the subscript on ?(r) gives the
In sample problem 39-7 we showed that the radial probability density for the ground state of the hydrogen atom is a maximum when r = a, where a is the Bohr radius, show that the average value of r,
An electron is confined to a narrow evacuated tube of length 3.0 m; the tube functions as a one-dimensional infinite potential well. (a) What is the energy difference between the electron's ground
As figure suggests, the probability density for the region x > L rn the finite potential well of Fig. 39-7 drops off exponentially according to ψ2(x) = Ce-2kx, where C is a constant. (a) Show
As figure suggests, the probability density for an electron in the region 0 2(x) = B sin2 kx, rn which B is a constant.? (a) Show that the wave function ?2(x) that may be found from this equation is
(a) For a given value of the principal quantum number n, how many values of the orbital quantum number ℓ are possible? (b) For a given value of ℓ how many values of the orbital magnetic
Let ∆Eadj be the energy difference between two adjacent energy levels for an electron trapped in a one-dimensional infinite potential well .Let E be the energy of either of the two levels. (a)
An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference AE between its quantum levels n and n + 2 is (h2/2mL2) (n + I).
Verify that the combined value of the constants appearing in Eq. 39-32 is 13.6 eV.
(a) Show that the terms in Schrodinger's equation (Eq. 39-18) have the same dimensions. (b) What is the common SI unit for each of these terms?
Repeat Sample Problem39-6 for the Balmer series of the hydrogen atom.
What is the probability that a state 0.0620eV above the Fermi energy will be occupied at (a) f = 0 K and (b) f = 320 K?
What is the number density of conduction electrons in gold, which is a monovalent metal? Use the molar mass and density provided in Appendix F.
(a) Show that Eq. 41-5 can be written as N(E) = CE1/2.(b) Evaluate C in terms of meters and electron-volts. (c) Calculate N(E) for E = 5.00eV.
Use Eq. 41-9 to verify 7.0eV as copper's Fermi energy.
Copper, a monovalent metal, has molar mass 63.54 g/mol and density 8.96 g/cm3. What is the number density n of conduction electrons in copper?
A state 63meV above the Fermi level has a probability of occupancy of 0.090. What is the probability of occupancy for a state 63meV below the Fermi level?
Show that Eq. 41-9 can be written as Ep: An2/3, where the constant A has the value 3.65 x 10-19 m2 ∙ eV.
Calculate the density of states N(E) for a metal at energy E = 8.0eV and show that your result is consistent with the curve of figure.
The Fermi energy for copper is 7.00 eV. For copper at 1000 K, (a) Find the energy of the energy level whose probability of being occupied by an electron is 0.900. For this energy, evaluate (b) The
Assume that the total volume of a metal sample is the sum of the volume occupied by the metal ions making up the lattice and the (separate) volume occupied by the conduction electrons. The density
In Eq. 41-6 let E – EF = ∆E = 1.00eV. (a) At what temperature does the result of using this equation differ by 10% from the result of using the classical Boltzmann equation P(E) =
What is the Fermi energy of gold (a monovalent metal with molar mass 197 g/mol and density 19.3 g/cm3)?
The Fermi energy for silver is 5.5 eV. At T = 0oC, what are the probabilities that states with the following energies are occupied: (a) 4.4 eV, (b) 5.4 eV, (c) 5.5 eV (d) 5.6 eV
What is the probability that an electron will jump across the energy gap Eg in a diamond that has a mass equal to the mass of Earth? Use the result of Sample Problem 41-1 and the molar mass of carbon
Calculate No (E), the density of occupied states, for copper at T = 1000 K for an energy E of(a) 4.00eV(b) 6.75 eY(c) 7.00 eV(d) 7.25 eV, and(e) 9.00 eV. Compare your results with the graph of
Show that the probability P(E) that an energy level having energy E is not occupied is P(E) = 1/e-∆E/KT + 1, where ∆E = E – Ep.
Silver is a monovalent metal. Calculate (a) The number density of conduction electrons, (b) The Fermi energy, (c) The Fermi speed, and (d) The de Broglie wavelength corresponding to this electron
At T = 300 K, how far above the Fermi energy is a state for which the probability of occupation by a conduction electron is 0.10?
The Fermi energy of aluminum is 11.6 eV its density and molar mass are 2.70 g/cm3 and 27.0 g/mol, respectively. From these data, determine the number of conduction electrons per atom.
Calculate the number density (number per unit volume) for (a) Molecules of oxygen gas at 0.0oC and 1.0 atm pressure and (b) Conduction electrons in copper. (c) What is the ratio of the
Show that, at T = 0 K, the average-energy E avg of the conduction electrons in a metal is equal to 3/5 EF.
A sample of a certain metal has a volume of 4.0 x 10–5 m3. The metal has a density of 9.0 g/cm3 and a molar mass of 60 g/mol. The atoms are bivalent. How many conduction electrons (or valence
Use the result of Problem 21 to calculate the total translational kinetic energy of the conduction electrons in 1.00 cm3 of copper at T = 0 K.
What is the number of occupied states in the energy range of 0.0300 eV that is centered at a height of 6.10eV in the valence band if the sample volume is 5.00 x 10-8 m3, the Fermi level is 5.00 eV
Zinc is a bivalent metal. Calculate (a) The number density of conduction electrons, (b) The Fermi energy, (c) The Fermi speed, and (d) The de Broglie wavelength corresponding to this electron speed.
A certain material has a molar mass of 20.0 g/mol, a Fermi energy of 5.00eV, and 2 valence electrons per atom. What is the density (g/cm3)?
At 1000 K, the fraction of the conduction electrons in a metal that have energies greater than the Fermi energy is equal to the area under the curve of Figure beyond EF divided by the area under the
At what temperature do 1.30% of the conduction electrons in lithium (a metal) have energies greater than the Fermi energy Ep, which is 4.70 eV?
(a) Using the result of Problem 21 and 7.00 eV for copper's Fermi energy, determine how much energy would be released by the conduction electrons in a copper coin with mass 3.10 g if we could
A certain metal has 1.70 x 1028 conduction electrons per cubic meter. A certain sample of that metal has a volume of 6.00 x 10–6 m3 and a temperature of 200 K. Flow many occupied states are in the
(a) What maximum light wavelength will excite an electron in the valence band of diamond to the conduction band? The energy gap is 5.50 eV. (b) In what part of the electromagnetic spectrum does
In a simplified model of an un-doped semiconductor, the actual distribution of energy states may be replaced by one in which there are Nv, states in the valence band, all these states having
The occupancy probability function (Eq. 41-6) can be applied to semiconductors as well as to metals. In semiconductors the Fermi energy is close to the midpoint of the gap between the valence band
The compound gallium arsenide is a commonly used semiconductor, having an energy gap E, of 1.43 eV. Its crystal structure is like that of silicon, except that half the silicon atoms are replaced by
Doping changes the Fermi energy of a semiconductor. Consider silicon, with a gap of 1.11 eV between the top of the valence band and the bottom of the conduction band. At 300 K the Fermi level of the
Pure silicon at room temperature has an electron number density in the conduction band of about 5 x 1015 m–3 and an equal density of holes in the valence band. Suppose that one of every 107 silicon
What mass of phosphorus is needed to dope l.0 g of silicon to the extent described in Sample Problem 41-6?
A silicon sample is doped with atoms having donor states 0.110 eV below the bottom of the conduction band. (The energy gap in silicon is 1.11 eV.) If each of these donor states is occupied with a
When a photon enters the depletion zone of a p-n-junction, the photon can scatter from the valence electrons there, transferring part of its energy to each electron, which then jumps to the
For an ideal p-n-junction rectifier with a sharp boundary between its two semiconducting sides, the current I is related to the potential difference V across the rectifier by I = I0 (eeV/KT = 1),
In a particular crystal, the highest occupied band is full. The crystal is transparent to light of wavelengths longer than 295 nm but opaque at shorter wavelengths. Calculate, in electron-volts, the
A potassium chloride crystal has an energy band gap of 7.6 eV above the topmost occupied band, which is full. Is this crystal opaque or transparent to light of wavelength 140 nm?
A certain computer chip that is about the size of a postage stamp (2.54 cm x 2.22 cm) contains about 3.5 million transistors. If the transistors are square, what must be their maximum dimension?
A silicon-based MOSFET has a square gate 0.50μm on edge. The insulating silicon oxide layer that separates the gate from the p-type substrate is 0.20μm thick and has a dielectric constant
(a) Find the angle θ between adjacent nearest-neighbor bonds in the silicon lattice. Recall that each silicon atom is bonded to four of its nearest neighbors. The four neighbors form a regular
Silver melts at 961oC. At the melting point, what fraction of the conduction electrons is in states with energies greater than the Fermi energy of 5.5 eV?
(a) Show that the density of states at the Fermi energy is given by in which n is the number density of conduction electrons. (b) Calculate N (EF) for copper, which is a monovalent metal with molar
Calculate dp/dT at room temperature for(a) Copper and(b) Silicon, using data from Table41-1.
(a) Show that the slope dP/dE of Eq. 41-6 at E = EF is –1/4kT. (b) Show that the tangent line to the curve of figure b at E = EF intercepts the horizontal axis at E = EF + 2kT.
Show that P (E)., the occupancy probability in Eq. 41-6, is symmetrical about the value of the Fermi energy; that is, show that P(EF + ∆E) + P(EF – ∆E) = 1.
At what pressure, in atmospheres, would the number of molecules per unit volume in an ideal gas be equal to the number density of the conduction electrons in copper, with both gas and copper at
Verify the numerical factor 0.121 in Eq. 41-9.
The Fermi energy of copper is 7.0eV. Verify that the corresponding Fermi speed is 1600 km/s.
Prove [AB, CD] = – AC {D, B} + A {C, B} D – C {D, A} B + {C, A} DB.
Suppose a 2 x 2 matrix X (not necessarily Hermitian. nor unitary) is written as X = α0 + σ ∙ a, where α0 and α1, 2, 3 are numbers.a. How are α0 and αk (k = 1, 2, 3) related to tr (X) and tr
Show that the determinant of a 2 x 2 matrix ? ? a is invariant under Find ?'k in terms of ?'k when ñ is in the positive z-direction and interpret your result.
Prove [AB, CD] = – AC {D, B} + A {C, B} D – C {D, A} B + {C, A} DB. Discuss.
a. Consider two kets | α > and | β >. Suppose < α’| α>, < α”| α), and < α’| β>, < α”|β), are all known, where | α’>, | α”), form a complete set of
Suppose | i > and | j > are eigenkets of some Hermitian operator A. Under what condition can we conclude that | i > + | j > is also an eigenket of A? Justify your answer.
Using the ortho normality of | +) and | - >, prove [Si, Sj] = iεijkhSk, {Si, Sj} = (h2/2) δij, where Sx = h/2 (| +> < – | + |– > < + –),Sy = ih/2 (– | + > < – | + |
Construct |S ? n; +) such that S ? n | S ? n; + > = (h/2) |S ? n; +> where ? is characterized by the angles shown in the figure. Express your answer as a linear combination of | + > and |?
The Hamiltonian operator for a two-state system is given by, H = α (| l> <1 | – |2 > < 2| + |1 > < 2| + |2 > < 1|), where a is a number with the dimension of energy. Find
A two-state system is characterized by the Hamiltonian H = H11 |1> <1| + H22|2> <2| + H12 [|1> < 2| + |2> <1|], where H11, H22, and H12 are real numbers with the dimension of
A spin ½ system is known to be in an Eigen state of S ∙ n with Eigen value h/2, where n is a unit vector lying in the xz-plane that makes an angle γ with the positive z-axis.a. Suppose Sx is
A certain observable in quantum mechanics has a 3 x 3 matrix representation as follows: a. Find the normalized eigenvectors of this observable and the corresponding eigenvalues. Is there any
Let A and B be oversvables, suppose the simultaneous eigenkets of A and B {|α’, b’>} form a complete orthonormal set of base kets. Can we always conclude that [A, B] = 0? If your answer is
Two Hermitian operator anti commute {A, B} = AB + BA = 0. In it possible to have a simultaneous (that is, common) eigenket of A and B? Prove or illustrate your assertion.
Two observables A1 and A2, which do not involve time explicitly, are known not to commute, [A1, A3] ≠ 0, yet we also know that A1 and A2 both commute with the Hamiltonian: [A1, H] = 0, [A2, H] = 0.
a. The simplest way to derive the Schwarz inequality goes as follows. First, observe ( + ? |? >) ? 0 for any complex number A; then choose A in such a way that the preceding inequality reduces to
Find the linear combination of | + > and | – > kets that maximizes the uncertainty product uncertainty product <(∆Sx)2> <(∆Sy)2>. Verify explicitly that for the linear
Consider a three-dimensional ket space. If a certain set of ortho-normal kets?say, | 1 >, | 2 >, and | 1 > ? are used as the base kets, the operators A and B are represented by with a and b
a. Prove that (1/√2) (1 + iσx) acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle – π/2. (The minus sign signifies
Some authors define an operator to be real when every member of its matrix elements } basis in this case). Is this concept representation independent, that is, do the matrix elements remain real even
Construct the transformation matrix that connects the Sz diagonal basis to the Sx diagonal basis. Show that your result is consistent with the generalrelation
a. Suppose that f(A) is a function of a Hermitian operator A with the property A| α’> = α’| α’>. Evaluate < b’’| (A) | b’> when the transformation matrix from the α’
a. Let x and px be the coordinate and linear momentum in one dimension. Evaluate the classical Poisson bracket [x, F(px)] classical.b. Let x and px be the corresponding quantum-mechanical operators
a. Gottfried (1966) states that can be ?easily derived? from the fundamental commutation relations for all functions of F and G that can be expressed as power series in their arguments. Verify this
The translation operator for a finite (spatial) displacement is given by where p is the momentum operator. a. Evaluate b. Using (a) (or otherwise), demonstrate how the expectation value 4(x) changes
In the main text we discussed the effect of T (dx’) on the position and momentum eigenkets and on a more general state ket | α >. We can also study the behavior of expectation values < x
a. Verify (1.7.39a) and (1.7.39b) for the expectation value of p and p2 from the Gaussian wave packet (1.7.35).b. Evaluate the expectation value of p and p2 using the momentum-space wave
Consider a ket space spanned by the eigenkets {| ?? >} of a Hermitian operator A. There is no degeneracy. a. Prove that is the null operator, b. what is the significance of? c. Illustrate (a) and
Consider the spin-precession problem discussed in the text. It can also be solved in the Heisenberg picture. Using the Hamiltonian H = – (eB/mc) Sz = wSz, write the Heisenberg equations of motion

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