The wavefunction ψ for an electron confined to move within a box of linear size L = m, is a standing wave as shown.

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1 1. This question is about quantum aspects of the electron. The wavefunction ψ for an electron confined to move within a box of linear size L = m, is a standing wave as shown. State what is meant by a wavefunction. (b) State the position near which this electron is most likely to be found. (c) Calculate the momentum of the electron. IB Questionbank Physics 1

2 (d) The energy, in joules, of the electron in a hydrogen atom, is given by E = n where n is a positive integer. Calculate the wavelength of the photon emitted in a transition from the first excited state of hydrogen to the ground state. 18 (3) (e) The electron stays in the first excited state of hydrogen for a time of approximately t = s. (i) Determine the uncertainty in the energy of the electron in the first excited state. (ii) Suggest, with reference to your answer to (e)(i), why the photons emitted in transitions from the first excited state of hydrogen to the ground state will, in fact, have a small range of wavelengths. IB Questionbank Physics 2

3 (f) Diagram 1 shows the three lowest energy levels for an electron in the hydrogen atom. Using the energy axis on diagram 2, draw the three lowest energy levels for the electron in a box model. You do not have to put any numbers on the vertical axis. (Total 13 marks) 2. This question is about the de Broglie hypothesis. Describe the de Broglie hypothesis. IB Questionbank Physics 3

4 (b) A beam of electrons is accelerated from rest through a potential difference of 85 V. Show that the de Broglie wavelength associated with the electrons in the beam is 0.13 nm. (3) (c) Electrons with the same kinetic energy as those in (b) are incident on a circular aperture of diameter 1.1 nm. The electrons are detected beyond the aperture. IB Questionbank Physics 4

5 The graph shows the variation with angle θ of the number n of electrons detected per second after diffraction by the aperture. Use your answer to (b) above to explain how data from the graph support the de Broglie hypothesis. (4) (Total 9 marks) IB Questionbank Physics 5

6 3. This question is about electrons. In 1924, Davisson and Germer carried out an experiment in which electrons were accelerated through a potential difference of 54 V. The electrons were scattered at the surface of a nickel crystal. (i) Outline how the results of the experiment suggested that electrons exhibit wave properties. (ii) Calculate the de Broglie wavelength of the electrons. (3) IB Questionbank Physics 6

7 (b) Explain how the de Broglie hypothesis is used with the electron in a box model to understand the origin of atomic energy levels in the atom. (5) (Total 10 marks) IB Questionbank Physics 7

8 4. This question is about atomic and nuclear spectra. Atomic spectra The diagram represents some of the energy levels of the mercury atom. Photons are emitted by electron transitions between the levels. On the diagram draw arrows to represent the transition, for those energy levels that gives rise to, (i) (ii) the longest wavelength photon (label this L). the shortest wavelength photon (label this S). (b) Determine the wavelength associated with the arrow you have labelled S. (3) IB Questionbank Physics 8

9 Nuclear spectra (c) A nucleus of the isotope bismuth-212 undergoes α-decay into a nucleus of an isotope of thallium. A γ-ray photon is also emitted. Draw a labelled nuclear energy level diagram for this decay. (d) The activity of a freshly prepared sample of bismuth-212 is Bq. After 80.0 minutes the activity is Bq. Determine the half-life of bismuth-212. (4) (Total 11 marks) IB Questionbank Physics 9

10 5. This question is about the hydrogen atom. A parallel beam of visible light is shone through monatomic hydrogen gas. The radiation emerging from the gas is analysed by comparing the incident and emerging intensities at various wavelengths. It is found that at a wavelength of 490 nm the intensity of the emergent beam is greatly reduced. The diagram shows some of the electron energy states of the hydrogen atom where n is the quantum number of the energy level. (i) Calculate the energy, in ev, of a photon of light of wavelength 490 nm. IB Questionbank Physics 10

11 (ii) Use your answer in (i) and the energy level diagram to explain the reduction in intensity of the emergent beam. (4) (b) Outline how the Schrödinger model of the hydrogen atom leads to the idea of discrete electron energy states. (4) (Total 10 marks) IB Questionbank Physics 11

12 6. This question is about the wave nature of matter and quantum energy states. Describe what is meant by the de Broglie hypothesis. (b) An electron is confined to one dimension in a box of length L. The de Broglie waves associated with the particle form standing waves in the box with wavelengths given by 2L where n is = 1, 2, 3, etc. n Show that the energy levels E n for the particle are given by E n = Planck s constant. n 2 h 2 2 (8mL ) where h is (3) (c) The electron makes a transition from the energy state given by n = 4 to n = 2. The length L = m. Calculate the (i) energy of the photon emitted. IB Questionbank Physics 12

13 (ii) wavelength of the photon emitted. (Total 9 marks) 7. This question is about the de Broglie hypothesis. State the de Broglie hypothesis. (b) Determine the de Broglie wavelength of a proton that has been accelerated from rest through a potential difference of 1.2 kv. IB Questionbank Physics 13

14 (c) Explain why a precise knowledge of the de Broglie wavelength of the proton implies that its position cannot be observed. (Total 6 marks) 8. This question is about α-particle scattering and nuclear processes. Radium-226 decays with the emission of α-particles to radon (Rn). Complete the nuclear reaction equation Ra Rn + (b) The decay constant of radium-226 is s 1 and each emitted α-particle has an energy of J. (i) Calculate the half-life of radium-226. IB Questionbank Physics 14

15 (ii) Determine the rate, in watts, of emission of energy from 1.0 g of radium-226. (4) (c) Experimental evidence that supports a nuclear model of the atom was provided by α-particle scattering. The diagram represents the path of an α-particle as it approaches and then recedes from a stationary gold nucleus. (i) On the diagram, draw lines to show the angle of deviation of the α-particle. Label this angle D. (ii) The gold nucleus is replaced by another gold nucleus that has a larger nucleon number. Suggest and explain the change, if any, in the angle D of an α-particle with the same energy and following the same initial path as in (c)(i). IB Questionbank Physics 15

16 (d) Estimate the distance of closest approach to a gold nucleus (Z = 79) of an α-particle with an initial kinetic energy of 4.0 MeV. (3) (Total 13 marks) 9. This question is about nuclear energy levels and radioactive decay. The diagram shows some of the nuclear energy levels of the boron isotope 12 5 B and the carbon 12 isotope 6 C. Differences in energy between the levels are indicated on the diagram. A particular beta decay of boron and a gamma decay of carbon are marked on the diagram. Calculate the wavelength of the photon emitted in the gamma decay. IB Questionbank Physics 16

17 (b) Calculate the maximum kinetic energy of the electron emitted in the beta decay indicated. (c) Explain why the electrons emitted in the indicated beta decay of boron do not always have the kinetic energy calculated in (b). (Total 5 marks) 10. This question is about fundamental interactions. State an exchange particle for (i) the weak interaction. (ii) the electromagnetic interaction. (b) Comment, with reference to the mass of the exchange particles, on the range of the weak and electromagnetic interactions. IB Questionbank Physics 17

18 (c) Describe the process represented by the Feynman diagram below. (d) State what is meant by a virtual particle. (e) Explain how the Heisenberg uncertainty principle for energy and time applies to the interaction in (c). IB Questionbank Physics 18

19 (f) The uncertainty in the time for the electromagnetic interaction between two electrons is s. Determine the uncertainty in the energy of the virtual photon. (Total 10 marks) IB Questionbank Physics 19

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