PHY2002 UNIVERSITY OF EXETER SCHOOL OF PHYSICS JANUARY 2010 QUANTUM PHYSICS I. Hints and Tips

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1 PHY2002 UNIVERSITY OF EXETER SCHOOL OF PHYSICS JANUARY 2010 QUANTUM PHYSICS I Hints and Tips 1. When you substitute the first and second spatial derivatives into the Schrödinger equation, note that the equation you find should be true for all values of x. 2. Check that you are familiar with the procedure for normalization and probability calculations from the worked examples in class. 3. (i) As a start consider what boundary conditions need to be satisfied. Are they in this example? (ii) Again, consider the boundary conditions. 4. The key here is to use the ideas of interference for the paths that the neutrons could take. 5. Again, this really comes down to applying boundary conditions on the wavefunction. Watch out for degeneracy 1 PHY2002

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11 PHYSICS EXAMINATION PROBLEMS SOLUTIONS AND HINTS FOR STUDENT SELF-STUDY Module Code Name of module PHY2018 MATHEMATICS WITH PHYSICAL APPLICATIONS Date of examination June (i) y = c exp( 4 x) + dx exp( 4x); y c = 6 sin( 2x) + 8 cos( 2x) (ii) (iii) y = 1 3 x 4 + cx u( x, y) = 4exp ("x + 3y 2) 2. 1-d wave-equation is "2 u "t 2 = #" 2 u& a2 % $ "x 2 ( where u is displacement. String is released from rest so ' "u "t = 0 at t = 0. Fourier coefficients are a n = 2 $ x( 1" x)sin( n#x)dx. 3. Inside: V r," ( ), E = "3V 0 cos# r ˆ + 3V 0 sin# # ˆ # % r 1+ 3 $ r cos (") & (, E = V 0 # 6cos" & 1+ ' 2 % ( r ˆ + 3 V 0 r $ r ' r 2 sin" " ˆ. ( ) = V rcos (") Outside: V ( r," ) = V 0 # 2 0 1& % ( 4. " = % ( $ 0 1 2' " = 3 " )1 = 1 # 1 1 )1& % ( )2 4 )1 3 % ( $ 1 )2 2 ' # 0& % ( H = 1 % ( $ 1' * = 0.34 rad 5. " y 1 % "( % " $ ' = $ ' x 1% $ ' # y 2 & # (0.02&# x 2 & " eignvalues: " 1 = 0 " 2 = #0.05 eigenvectors: +2 % " +1% $ ', $ ' # +3& #(1& at t = 60min : y 1 ~ 61, y 2 ~ E 0 (1) = 3"h2 4m 2 # 2 1

12 PHY2019 Observing the Universe: Hints and Tips for June 2010 Examination 1. Minimum binary separation resolvable: use Rayleigh Criterion and relation between angular separation and physical separation Number of photons per second incident on HST s collecting area: [ photons s 1 ] Hint: convert magnitude to flux and calculate total energy incident on telescope per second (flux collecting area). How many photons per second is this? (Assume all photons have λ = 550 nm.) Signal-to-noise ratio (S/N) after 2 minute observation: [ S/N 1600 ] Hint: number of photoelectrons detected = number of incident photons per second efficiency factor integration time. The uncertainty (noise) comes from N fluctuations in the photo-electron count. Limiting magnitude for binary compansions: [ V = 26.0 ] Hint: S/N ratio of 10 requires 100 photo-electrons. Calculate photon flux required, convert to energy flux and then to equivalent magnitude. 2. Planck Functions: basic definitions and knowledge. Rayleigh-Jeans approximation: expand Planck Function for hν/kt << 1. Luminosity of Sun: L =4πR 2 σt 4. Equilibrium temperature of Earth: Hint: balance energy absorbed by Earth (over cross sectional area πr 2 E ) and energy re-radiated by Earth (over total area 4πR2 E ). Equilibrium temperature of Kuiper Belt objects: Hint: same calculation as Earth; expedient to express everything in convenient units... Herschel observations of Kuiper Belt Objects: Hint: use Wien s Law. Space telescope? Earth s atmosphere is opaque to this radiation. Discovery of Kuiper Belt objects optically: Hint: some light is reflected. 3. (i) Kepler I and II: basic knowledge. Kepler III: equate gravitational force with centripetal force required for circular orbit and rearrange. (ii) Methods for extrasolar planet detection: basic knowledge. Mass of orbiting object: [ m 2 = kg] Hint: use relation between velocity semiamplitude and mass function, suitably modified in the limit m 2 << m 1. This is a lower limit as the orbit may be inclined at i<90. Dip in lightcurve: eclipsing system; period confirmed; i = 90; m 2 accurately derived. Fractional dip in flux equal to proportion of stellar disk eclipsed by planet (assume planet contributes negligible flux). Object s orbital radius: simple application of Kepler s 3rd Law.

13 4. (i) Free-fall time: simple derivation given in lectures. Turbulence/magentic field: basic knowledge. (ii) Protostars and star formation process: basic knowledge. 5. Proper distance/coordinate distance: basic knowledge. Hubble Law: straightforward differentiation. Cosmlogical redshift: basic knowledge. Temperature of CMB when emitted: Hint: cosmological redshift gives ratio of observed and emitted wavelengths. Combine this with Wien s Law. Early universe: basic knowledge.

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16 PHYSICS EXAMINATION PROBLEMS SOLUTIONS AND HINTS FOR STUDENT SELF-STUDY Module Code Name of module PHY2208 OPTICS Date of examination June (i) (a) 15 cm behind mirror, size 3 cm. (b) cm behind mirror, size 4.51 cm. 2. (ii) (a) 45, (b) 15, (c) h = 0.075L 3. Write the (Fourier Transform) relation for the Spectral intensity I(k) in terms of the given photoelectric intensity as a function of mirror separation I(D), express cos() in terms of complex exp() functions: I 0 ( k) " # # 2 + 4k 2 4. (i) (b) " = 2n# kd for n = ±1,±2,±3K (ii) W " S# kd 5. Not applicable 1