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physics
modern physics

Questions and Answers of Modern Physics

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths
The contour representation of one of the orbital's for the n = 3 shell of a hydrogen atom is shown below.(a) What is the quantum number l for this orbital?(b) How do we label this orbital?(c) How
The watt is the derived SI unit of power, the measure of energy per unit time: 1W = 1J-s. A semiconductor laser in a CD player has an output wavelength of 780 nm and a power level of 0.10 m W. How
Carotenoids are yellow, orange, and red pigments synthesized by plants. The observed color of an object is not the color of light it absorbs but rather the complementary color, as described by a
A photocell is a device used to measure the intensity of light. In a certain experiment, when light of wavelength 630 nm is directed onto the photocell, electrons are emitted at the rate of 2.6 x
In an experiment to study the photoelectric effect, a scientist measures the kinetic energy of ejected electrons as a function of the frequency of radiation hitting a metal surface. She obtains the
The human retina has three types of receptor cones, each sensitive to a different range of wavelengths of visible light, as shown in this figure (the colors are merely to differentiate the three
The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are
When the spectrum of light from the Sun is examined in high resolution in an experiment similar to that illustrated in Figure 6.9, dark lines are evident. These are called Fraunhofer lines, after the
Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as He+ and Li2+.(a) Why is the Bohr model applicable to He+ ions but not to neutral He atoms?(b) The
An electron is accelerated through an electric potential to a kinetic energy of 18.6 ke V. What is its characteristic wavelength?
In the television series Star Trek, the transporter beam is a device used to "beam down" people from the Starship Enterprise to another location, such as the surface of a planet. The writers of the
The drawing below shows part of the orbital diagram for an element.(a) As drawn, the drawing is incorrect. Why?(b) How would you correct the drawing without changing the number of electrons?(c) To
Consider the discussion of radial probability functions in "A Closer Look" in Section 6.6. (a) What is the difference between the probability density as a function of r and the radial probability
For orbital that are symmetric but not spherical, the contour representations (as in Figures 6.22 and 6.23) suggest where nodal planes exist (that is, where the electron density is zero). For
The "Chemistry and Life" box in Section 6.7 described the techniques called NMR and MRI. (a) Instruments for obtaining MRI data are typically labeled with a frequency, such as 600 MHz. Why do you
Suppose that the spin quantum number, ms, could have three allowed values instead of two. How would this affect the number of elements in the first four rows of the periodic table?
Using the periodic table as a guide, write the condensed electron configuration and determine the number of unpaired electrons for the ground state of (a) Si, (b) Zn, (c) Zr, (d) Sn, (e) Ba, (f) Tl.
Scientists have speculated that element 126 might have a moderate stability, allowing it to be synthesized and characterized. Predict what the condensed electron configuration of this element might
Microwave ovens use microwave radiation to heat food. The energy of the microwaves is absorbed by water molecules in food and then transferred to other components of the food. (a) Suppose that the
The stratospheric ozone (O3) layer helps to protect us from harmful ultraviolet radiation. It does so by absorbing ultraviolet light and falling apart into an O2 molecule and an oxygen atom, a
The discovery of hafnium, element number 72, provided a controversial episode in chemistry. G. Urbain, a French chemist, claimed in 1911 to have isolated an element number 72 from a sample of rare
In a 1H NMR spectrum the peak from TMS is found to occur at 500.134 271 MHz. Two other peaks in the spectrum are found at 500.135 021 and 500.137 921 MHz; compute the chemical shifts of these two
Following the approach described in section 2.3.1 on page 10, use a tree diagram to predict the form of the multiplet expected for spin A when it is coupled to two other spins, B and C, with coupling
A rotation has a period of 2.5 x 10-9 s; compute the corresponding frequency in both Hz and rad s-1. Compute how long it will take the object to rotate through an angle of: (a) 90°; (b) 3π/2
Following the style of Fig. 2.17 on page 18, make sketch graphs of the x- and y-components of a rotating particle as a function of time for the case where the starting phase < p is: (a) 0°. (b)
The y-component is r sin (ωt + π). Using the same approach as on page 19, show that this y-component can be written more simply as -r sin (ωt).
Following the same approach used in section 3.2.7 on page 30, show that ψ-1/2 is an eigenfunction of one spin with eigenvalue
Calculate the Larmor frequency (in Hz, MHz and rad s-1) of l3C at a magnetic field strength of 9.4 T; the gyromagnetic ratio of l3C is +6.7283 x 107 rad s-l T-l and you assume that the chemical shift
Show that Ïˆ±1/2 are eigenfunctions of the one-spin Hamiltonian when it is written in angular frequency unitshence find the corresponding eigenvalues. You should assume thati.e. that Ïˆ±1/2,
Following the approach of section 3.5 on page 35, show that Ïˆa.1 Ïˆa.2 is an eigenfunction of the Hamiltonian for two spins with no coupling between them:Hence find the
Larmor frequencies are usually tens or hundreds of MHz, but to make the numbers easier to handle in this problem we will assume that the Larmor frequencies are very much smaller. Consider a system
For a three-spin system, use Table 3.3 on page 41 to work out the frequencies of the four allowed transitions in which spin two flips. Then, taking v0,2 = -200 Hz, 723 = 4 Hz and the rest of the
For a three-spin system, use Table 3.3 on page 41 to compute the frequencies of the six zero-quantum transitions; mark these transitions on an energy level diagram. Explain why these six transitions
A particular spectrometer has a B0 field which gives a Larmor frequency of 600 MHz for 1H; the RF field strength, ω1/(2π), has been determined to be 25 kHz. Suppose that the transmitter is placed
Use vector diagrams to predict the outcome of the sequence: 90°(x) - τ - 90°(x) when applied to equilibrium magnetization. In your answer, explain how the x-, y- and z magnetizations depend on the
Consider the spin echo sequence to which a 90° pulse has been added at the end: 90°(x) - τ - 180°(x) - τ - 90°(ϕ). The axis about which the pulse is applied is given in brackets after the flip
The so-called l- sequence is: 90°(x) - τ - 90°(ϕ) For a peak which is on resonance the sequence does not excite any observable magnetization. However, for a peak with an offset such that Ωτ =
The so-called 1-1 sequence is: 90°(x) - τ - 90°(y). Describe the excitation that this sequence produces as a function of offset. How could it be used for observing spectra in the presence of
If there are two peaks in the spectrum, we can work out the effect of a pulse sequence by treating the two lines separately. There is a separate magnetization vector for each line. Suppose that a
Explain why it is that the maximum signal in a pulse-acquire experiment is seen when the flip angle of the pulse is 90°. What would you expect to see in such an experiment if the flip angle of the
In an experiment to determine the pulse length, an operator observed a positive signal for pulse widths of 5 and 10 μs; as the pulse was lengthened further the intensity decreased going through a
Using an approach similar to that of Fig. 4.24 on page 64, show that a 180° poise about the y-axis rotates vectors to mirror image positions with respect to the yz-plane. [As in this figure, resolve
Use vector diagrams, similar to those of Fig. 4.25 on page 65, to show what happens during the spin echo sequence 90°(x) - τ - 180°(y) - τ - Also, draw up a phase evolution diagram appropriate
The gyromagnetic ratio of 31P is 1.08 x 108 rad s-1 T-1. This nucleus shows a wide range of shifts, covering some 700 ppm. Estimate the minimum 90° pulse length you would need to excite peaks over
A spectrometer operates at a Larmor frequency of 400 MHz for 1H and hence 100 MHz for l3C. Suppose that a 90° pulse of length 10 μs is applied to the protons. Does this have a significant effect on
From the plots of Fig. 4.28 on page 68 we see that there are some offsets at which the transverse magnetization goes to zero. Recall that during the pulse the magnetization starts on +z and is
When calibrating a pulse by looking for the null produced by a 180° rotation, why is it important to choose a line which is close to the transmitter frequency (i.e. one with a small offset)?
In a spectrum with just one line, the dispersion mode lineshape might be acceptable - in fact we can think of reasons why it might even be desirable (what might these be?). However, in a spectrum
Explain why use of a sine bell weighting function shifted by 45° may enhance the resolution of a spectrum, whereas use of a sine bell shifted by 90° will not.
In a proton NMR spectrum the peak from TMS was found to show 'wiggles' characteristic of truncation of the FID. However, the other peaks in the spectrum showed no such artefacts. Explain.
An absorption mode Lorentzian line, centred at frequency Î© is given byWe are going to work out the width of the line at half-height. Clearly, this is independent of the position of the
A dispersion Lorentzian line, centred at Î© = 0, is given byTo find the minima and maxima, we first need to differentiate D(Ï‰) with respect to co; show that this differential
Make sketches similar to those of Fig. 5.9 on page 87 for the cases: (a) ϕ = 3π/4. (b) ϕ = 3/r/2. (c) ϕ = 2π. (d) ϕ = 5π/2.
Suppose that we record a spectrum with the simple pulse-acquire sequence using a 90° pulse applied along the x-axis. The resulting FID is Fourier transformed and the spectrum is phased to give an
The gyromagnetic ratio of 31P is 1.08 x 108 rad s-1 T-1. This nucleus shows a wide range of shifts, covering some 700 ppm. Suppose that the transmitter is placed in the middle of the shift range and
Why is it undesirable to continue to acquire the time-domain signal after the NMR signal has decayed away? How can weighting functions be used to improve the SNR of a spectrum? In your answer
For a time-domain signal in which the NMR signal has decayed long before the end of the acquisition time, explain why the SNR of the corresponding spectrum can be improved either by shortening the
Describe how weighting functions can be used to improve the resolution in a spectrum. In practice, what sets the limit on the improvement that can be obtained? Will zero filling improve the
Write the following in Dirac notation:Express the following statements in Dirac notation: (a) Ïˆa is normalized; (b) Ïˆa and ÏˆÎ² are orthogonal; (c)
If the wavefunction for a single spin is given byand assuming that (c*ca + c*Î²c*Î²) = l, show that What is the interpretation of ?
Using the approach of section 6.3.5 on page 111, show that
Given thatand expressing the coefficients in r/Ï• format: show that Use the identity exp (iÎ¸) - exp (-iÎ¸) = 2i sin Î¸ to tidy this up to Hence show
Using the approach of section 6.6.1 on page 119, show that[Start from Eq. 6.31 on page 120 and left multiply by (Î²)]
Using the approach of section 6.6.3 on page 122, show that, during a period of free evolution, the expectation value of y evolves according toGive a graphical interpretation of this result.
Using the approach described in section 6.8.1 on page 127, show that for a spin described by the wavefunctionthe matrix representation of the density operator is given by
Using Fig. 7.4 on page 148, determine the result of the following rotations:Write each of these transformations using the arrow notation introduced in section 7.1.2 on page 141.
For a homonuclear two-spin system, what delay r in a spin-echo sequence would you use to achieve the following overall transformations? (Apart from in the last transformation, do not worry about the
Consider the spin-echo pulse sequence shown in Fig. 7.14 (c) on page 164. By considering the evolution of the operators x and 2zx, show that the coupling is refocused, and that, from the point of
Why does the second 90° pulse to spin one in the INEPT experiment (Fig. 7.15 on page 168) have to be about the y-axis?Show that changing the phase of the first 90° pulse from x to -x results
Specify the coherence order (or orders) of the following operators:
By expressing x and y in terms of + and _, verify the relationships given in the table on page 175.
Consider the pulse sequence shown below.Starting with equilibrium magnetization on spin one, 1z, show that the sequence generates a mixture of double- and zero-quantum coherence. Find the value of r
Show that
Following the same approach as in section 7.2.2 on page 145, show that (for a single spin) a spin-echo sequence in which the 180° pulse is about y 90o(x) - τ - 180°(y) - τ - results in the
Determine the outcome of the following rotations:
A variant on the 1 - sequence, described in section 7.3.3 on page 148, is the sequence: 90°(x) - τ - 90°(y). Show that, for a one-spin system, this sequence gives rise to trans-verse
Using Fig. 7.6 on page 152, determine the outcome of the following, all of which involve the evolution of coupling:
Following the approach of section 7.5 on page 152, show that the observable signal arising from 1y is of the formHence describe the spectrum you expect to see. Similarly, determine the observable
Assuming that magnetization along the y-axis gives rise to an absorption mode lineshape, draw sketches of the spectra which arise from the following operators:Describe each spectrum in words.
Give the outcome of the following evolution due to pulses or delays. In each case, describe the overall transformation in words.
Using the same approach as in section 7.8.1 on page 159, show that the effect of a spin echo (in a homonuclear system) on the terms 21x2z and 21yz is as follows:Using the idea that a spin echo is
Identify the preparation and mixing periods for each two-dimensional pulse sequence described in this chapter.
Discuss how you would modify the HSQC pulse sequence to make it suitable for detecting correlations through long-range l3C-1H couplings.
It was shown that for the TOCSY sequence (applied to a two-spin system) the observable signals at the start of t2 are given by Eq. 8.11 on page 223:Using the same approach as in section 8.3.2 on page
A peak in the P-type spectrum can be represented asThe real part of this phase-twist lineshape is plotted in Fig. 8.36 on page 229. What does the imaginary part look like? Sketch a contour plot.
Describe how you would apply (a) the SHR, and (b) the TPPI method to the HSQC sequence.
For the double-quantum experiment whose pulse sequence is shown in Fig. 8.1 8 on page 204, it turns out that shifting the phase of all the pulses which precede t\ by 0 results in the t\ modulation
The diagram below is the same as Fig. 8.5 (b) on page 187 i.e. the result of Fourier transforming the time-domain data along t2. Sketch the form of the cross-sections indicated by the arrows at
Starting with equilibrium magnetization on spin two, 2z, determine the form of the observable operators present at t2 = 0 in the COSY sequence. Describe the kinds of peaks (cross or diagonal) which
Repeat the previous exercise, but this time for the DQF COSY pulse sequence. Previous Exercise Starting with equilibrium magnetization on spin two, 2z, determine the form of the observable operators
A two-dimensional zero-quantum spectrum can be recorded using the same pulse sequence as for double-quantum spectroscopy, Fig. 8.18 on page 204. The only difference is that zero quantum coherence is
Why must the second I spin 90° pulse in the HSQC sequence be applied about the y-axis? What would the effect be of applying this pulse about the -y-axis?
For the HSQC sequence of Fig. 8.22 (a) on page 210, show that changing the phase of the first 90° pulse to the S spin from x to y results in the modulation of the observed signal changing from
For the HMQC sequence, as applied to proton and 13C, explain how difference spectroscopy can be used to suppress the contributions from protons which are not coupled to l3C. Make an explicit
Explain why, in an HMQC, the peaks will have their maximum intensity when τ = n/(2JIS) for n = 1, 3, 5 ..., and zero intensity for n = 2, 4, 6 ....
Suppose that an NMR sample containing 1013 protons is placed in a magnetic field of 9.4 T. Using the Boltzmann distribution, calculate the equilibrium populations of the α and β levels, assuming
Comment on the values you obtain at the two fields, and compare them with the dipolar contributions calculated in the previous question.[The values you should find at 4.7 T are c2 = 1.00 x I09 s-2,
Protons have much smaller chemical shift anisotropics than heteronuclei, but at the very high magnetic fields which are now becoming available CSA relaxation may be significant. Assuming that a
An alternative to the transient NOE experiment described in section 9.7.1 on page 274, is one in which rather than spin two being inverted by a selective 180° pulse at the start of the experiment,

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