Classical and Modern Physics
Classical and Modern Physics PHYS 211
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This 4 page Class Notes was uploaded by Ms. Janiya Heaney on Saturday October 3, 2015. The Class Notes belongs to PHYS 211 at Bucknell University taught by Staff in Fall. Since its upload, it has received 70 views. For similar materials see /class/218098/phys-211-bucknell-university in Physics 2 at Bucknell University.
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Date Created: 10/03/15
Some additional math and physics rules Series expansions An exponential function can be written as a series as follows x4 x5 x6 x7 x x2 x3 2 1x 2 3 4 5 6 7 Similarly the sine and cosine functions can be written as follows 3 S sin66 2 3 5 7 2 4 6 cos 1 6 9 2 4 6 The complex exponential In general em cos9isin 9 e49 cosH isinH Cosines and Sines can be written in terms of the complex exponential as i3 i9 e 6 C030 e sm 0 21 T ime dependent states Assume that a system is initially in the state Ill0 a1 1azl 2gtall 3 39 where the states 1gt M2 M3 are states with definite energies E E2 E3 Then state of the system at any arbitrary time t is given by Wtgta1e iElth1 lgta2e iE11hl 2a3e iE3th 3u This is important because systems change in time A system might be in a particular state at some time but that state will typically but not always change For instance if you locate an electron and find it to be at a certain location then immediately after that measurement you know the electron is there But it won t typically stay at that location nor will its location even remain determined The rule stated here although not derived shows how to determine how a system will change in time after a measurement All the other math and physics rules still hold You will use the same rules as before to determine probabilities and expectation values Magnetic moments in an external magnetic field Recall from earlier this semester that a current loop has a magnetic moment 1 the points in the direction of the magnetic field produced by the loop Experimentally it has been found that elementary particles typically have a magnetic moment that is aligned with the particles spin angular momentum The properties of magnetic moments discuSSed earlier in the semester apply here as well The attached sheet from Halliday Resnick and Walker discusses the issues of energy of a magnetic moment in an external magnetic field But there is also a torque acting on a magnetic moment Recall from earlier in our discussion of electricity and magnetism that the torque on a magnetic moment is given by f x If the magnetic moment is aligned with the external magnetic field or if they point in the opposite direction there is no torque acting on the moment If the angle between and E is not 0 or 180 then the torque is nonzero and will cause the magnetic moment to twist In fact it will cause the moment to twist toward the magnetic field However since the particle also has an angular momentum it will not simply swing toward the magnetic field Rather it will precess in a circle around the magnetic field much in the same way that a gyroscope precesses when in a gravitational field 39 meklg v ramsquot 6 Wm 1h NW emo 4Ky 4o6 Magnetic Resonance As we discussed brie y in Section 327 a proton has a spin magnetic dipole moment If that is associated with the proton s intrinsic spin angular momentum S The two vectors are said to be coupled together and because the proton is positively charged they are in the same direction Suppose a proton is located in a magnetic eld B that is directed along the positive direction of a z aXiSuThen has two possible quantized components along that axis the component can be n if the vector is in the direction of B Fig 40 10a or Lz if it is opposite the direction of B Fig 40401 From Eq 28 38 U0 II B recall that a potential energy is associated with the orientation of any magnetic dipole moment It located in an external V magnetic eld B Thus energy is associated with the two orientations shown in Figs 4010a and b The orientation in Fig 4010a is the lower energv state and is called the spinup state because the proton s spin component Sz not shown is also aligned with B The orientation in Fig 4010b the spindown state is the higherenergy state 173 Thus the energy difference between these two states is AE uzB MzB 21Lsz 4021 If we place a sample of water in a magnetic eld 1 3 the protons in the hy drogen portions of each water molecule tend to be in the lower energy state We shall not consider the oxygen portions Any one of these protons can jump to the higherenergy state by absorbing a photon with an energy hf equal to AE That is the proton can jump by absorbing a photon of energy hf anB 4amp22 Such absorption is called magnetic resonance or as originally nuclear magnetic resonance NMR an th 1 reversal of the spin S is called spinflipping In practice the photons required for magnetic resonance have an associated frequency in the radio frequency RF range and are provided by a small coil wrapped around the sample undergoing resonance An electromagnetic oscillator called an RF source drives a sinusoidal current in the coil at frequency f The electromagnetic EM eld set up within the coil and sample also oscillates at frequency f If f meets the requirement of Eq 40 22 the oscillating EM eld can transfer a quantum of energy to a proton in the sample via a photon absorption spin ipping the proton The magnetic eld magnitude B that appears in Eq 4022 is actually the magnitude of the net magnetic eld 1 5 at the site where a given proton undergoes spin ipping That net eld is the vector sum of the external eld Em set up by the magnetic resonance equipment primarily a large magnet and the internal eld Bin set up by the magnetic dipole moments of the atoms and nuclei near the given proton For practical reasons we do not discuss here magnetic reso nance is usually detected by sweeping the magnitude Bext through a range of values while the frequency fof the RF source is kept at a predetermined value and the energy of the RF source is monitored A graph of the energy loss of the RF source versus Bm shows a resonance peak when B sweeps through the value at which spin ipping occurs Such a graph is called a nuclear magnetic resonance spectrum or NMR spectrum Figure 4011 shows the NMR spectrum of ethanol which is a molecule con sisting of three groups of atoms CH3 CH2 and OH Protons in each group can undergo magnetic resonance but each group has its own unique magneticres onance value of B because the groups lie in different internal elds Bin due to their arrangement within the CH3CHZOH molecule Thus the resonance peaks 40 6 MagneticResonance 1121 a 1 B B y a b Spin down J E 2418 Spin up T E Fig quzu The 1 component of F for a proton in the a lowerenergy spinup and b higherAenergy spin down state c An energy level diagram for the states showing the upward quantum jump the pro ton makes when its spin ips from up to down Fig 4011 A nuclear magnetic resonance spectrum for ethanol CH3CHZOH The spectral lines rep resent the absorption of energy as sociated with spin ips of protons The three groups of lines corre spend as indicated to protons in the OH group the CH2 group and the CH3 group of the ethanol mole cule Note that the two protons in the CH2 group occupy four different local environments The entire hori zontal axis covers less than 10 4 T 1122 Chapter 40 All About Atoms Fig 4012 A cross sectional view of a human head and neck produced by magnetic resonance imaging Some of the details visible here would not show up on an x ray im age even with a modern computerized axial tomography scanner CAT scanner in the spectrum of Fig 4011 form a unique NMR signature by which ethanol can be indenti ed Because many substances have unique NMR signatures magnetic resonance is used to identify unknown substances such as in forensic work of a criminal investigation Additionally a procedure called magnetic resonance imaging MRI has been applied to medical diagnostics with great success The protons in the various tissues of the human body are situated in many different internal magnetic environments When the body or part of it is immersed in a strong external magnetic eld these environmental differences can be detected by spiri ip techniques and translated by computer processing into an image resembling those produced by x rays Figme 4012 for example shows a cross section of a human head imaged by this method In an NMR experiment a drop of water is suspended in a uniform external magnetic eld Em Assume the internal eld Him is negligible The magnitude of at for a proton in the hydrogen atoms of the water molecules is 141 X 10 26 JT Magnetic resonance occurs when Bem 180 T What is the frequency f of the RF source causing the protons to spin ip and what wavelength A is associated with a photon absorbed in the spin ipping Solution One Key idea here is that when a proton is located in an external magnetic eld EL it has a potential energy because it is a magnetic dipole A second Key Idea is that this potential energy is restricted to two values with a difference of ZuzB The third Key Idea is that if the proton is to jump between these two energies spin ip the photon energy hf must be equal to the energy difference ZMZB according to Eq 4022 From that equation with B Be 180 T we then nd 39 f 2113 2141 x 1026 JT180 T h 663x10 341 s 766 x 107 Hz 766 MHz Answer This is the frequency associated with the photons absorbed in the spin ipping it is also the frequency of the RF source and thus of the oscillating electromagnetic elds set up by that source The wavelength associated with a photon absorbed in the Spin ipping is c 300 gtlt108 ms 3 A f 766 X 107 Hz 92 to Answer