Page 1 of 5 PHY3101 3-29-00 Name:_______________________ Midterm 2 Closed book exam. One 8.5x11" study sheet is allowed. A calculator is allowed also. Exam is worth 30 points, 15% of your total grade. Forms of the Schrodinger equation: −∂∂+ =∂∂− + =hhh2 222 2222mxV x t itmddxV x Eψψψ ψψ ψ( , ) ( ) Constants and equations: h=×−662611034. J-s c=×30108.m/s e=×−160221019. C h=×−413571015. eV-s λ f = c 1 eV J=×−160221019. hc=1240 eV-nm f = E / h 1 MeV = 106 eV h = = ×−h / .2 10546 1034π J -s λ = h / p 1 nm = 109−m h = ×−65821 1016. eV-s 1 Watt = 1 J/s 1 MHz = 106s-1 me= ×−911 1031. kg mp= ×−1673 1027. kg mµ= ×−1881 1028. kg me=0511. MeV / c2 mp=9383. MeV / c2 mµ=1057. MeV / c2 e2094144 10πε= ×−. eV- m FEvB=+×q() WqV= REhc∞= = ×07109737 10. m-1 REhcH= = ×07109678 10. m-1 1 1 12 2λ= −FHGIKJRn nl u ame00221040 53 10= = ×−πεh. m EZ e mn nne=−= −2 4202 2 22 4136h ( ).πε eV Z2b g µBeem= = ×−h257884 105. eV / T ∆∆x p ≈h ∆∆E t ≈h $( )$( )*f x f x dx=zψ ψ x e dx n n n n nn x n−∞+z= = ⋅ − ⋅ − ⋅/! !αα011 2 2 1 bgbgL dx xxxsin sin22142z= − dx x xx xxx x 2 23 26 418224sin sincosz= − −FHGIKJ− I x x dxnn= −∞zexp α 20ch I012=πα I112=α I214=απαPage 2 of 5 PHY3101 3-29-00 Name:_______________________ 1. [3 points] If the wavelength of an electron is known to be 1 nm, what is its kinetic energy? 2. [3 points] If the lifetime of an atom in an excited state is 1 ns (10-9 s), what is the intrinsic uncertainty in the energy of the photon emitted in the transition to the ground state?Page 3 of 5 PHY3101 3-29-00 Name:_______________________ 3. Consider the step potential shown below. The energy E of a particle of mass m is less than the barrier height V0. V xV xx( )=>≤RSTUVW000 0 (a) [3 points] The solution to the time-independent Schrodinger Equation for region 1 (x<0) can be written ψ1( )x e R eikxikx= +− . What is the constant k? (b) [4 points] Solve the time-independent Schrodinger Equation to obtain the general solution for region 2 (x > 0). You should have at least one constant analogous to k in region 1 which depends on the barrier height, the particle energy, the particle mass, and h. (c) [3 points] Apply the boundary conditions at x =∞ and x = 0 to eliminate any constants and to solve the remaining constants in terms of R.V(x) x E V0 1 2 0Page 4 of 5 PHY3101 3-29-00 Name:_______________________ 4. The ground state wavefunction for the hydrogen atom is ψ0 0r A r aafbg= −exp / (a) [4 points] Determine the normalization constant A. You may neglect the angular integration, but remember that you must integrate over r dr2. (b) [4 points] Find the expectation value r . Again, remember that you must integrate over r dr2.Page 5 of 5 PHY3101 3-29-00 Name:_______________________ 5. [3 points] Consider an atom of hydrogen in the 3p state immersed in a 0.25 T magnetic field aligned along the z-axis. What is the energy splitting caused by the quantization of the z-component of the angular momentum? 6. [3 points] Explain what is meant by “tunneling” in Quantum