PHYS 30 Concept Tests Fall 009
In classical mechanics, given the state (i.e. position and velocity) of a particle at a certain time instant, the state of the particle at a later time A) cannot be determined B) is known more or less C) is uniquely determined
Is Einstein s eplanation of the photo effect enough to prove that energy quanta of radiation are particles? A) Yes B) No University of Colorado, Boulder (009)
Which new concept did Einstein use to eplain the photo effect? A) Light is a wave B) Light consists of energy quanta C) E = mc D) Light consists of particles E) I don t know University of Colorado, Boulder (009)
Compton did observe the effect using X-rays. Would he be able to detect the effect (change in wavelength) with visible light too? A) Yes, and I know why B) Yes, but I guess C) No, and I know why not D) No, but I guess University of Colorado, Boulder (009)
Compton effect: How many maima will be detected if the detector is placed in forward direction? A) 0 B) 1 C) D) 3 E) More University of Colorado, Boulder (009)
For an electron with energy 5 ev, λ ~.5 Angstrom. The slit separation in a standard double-slit eperiment with a scintillation screen is d ~ 0.5 mm. What do you epect to see? A) A nice interference pattern with minima and maima B) More or less a bright screen (no separation of minima and maima) C) A black screen, since the electrons are diffracted by large angles University of Colorado, Boulder (009)
Shown is a probability distribution P(j) for a discrete variable j. What is the average (mean) value <j>? P(j) 0.3 0.1-3 - -1 0 1 3 j A) <j> = - B) <j> = -1 C) <j> = 0 D) <j> = 1 E) <j> = University of Colorado, Boulder (009)
Shown is a probability distribution P() for a continous variable. What is the average (mean) value <>? P() 3/8 1/8-3 - -1 0 1 3 A) <> = - B) <> = -1 C) <> = 0 D) <> = 1 E) <> = University of Colorado, Boulder (009)
How does one measure an epectation value <>? A) Prepare an ensemble of many identical particles and measure the positions of each of the particle. B) Measure the position of the same particle again and again many times. University of Colorado, Boulder (009)
Requirements for the wave equation and its solution: 1. The equation has to be linear. The solution has to represent a frequency ω and wave vector k with ω=πe/h and k=πp/h 3. The solution must satisfy the dynamical equation E = p /m + V(r,t) Which of these requirement has to be fulfilled? A) None B) 1 and C) 1 and 3 D) and 3 E) all of them University of Colorado, Boulder (009)
Below is shown a probability distribution. At which point 0 is there equal probability for the particle to be either on the left or on the right of 0 P() A) 0 = - B) 0 = 0 C) 0 = 1 D) I cannot decide, but I know how to calculate it. University of Colorado, Boulder (009)
Ψ and ψ : Which of these quantities is measurable? A) None of them B) ψ C) ψ D) Both of them University of Colorado, Boulder (009)
Orbital = wavefunction tomography e - Eperiment N N Theory What is here wrong? University of Colorado, Boulder (008)
What do you read for PHYS30? A) Griffiths B) Lecture notes C) Griffiths and lecture notes D) More than Griffiths and lecture notes E) Nothing (besides notes taken in class) University of Colorado, Boulder (009)
What is? )) / ( ep( ), ( with ), ( ˆ )) / ( ep( ), ( with ), ( ˆ Et k i t t p Et k i t t p k k k k University of Colorado, Boulder (009) A) B) C) D) E) 0 ), ( ˆ ), ( ˆ t p t p k k k k k k k p k p ˆ and ˆ k k k k k p k p ˆ and ˆ k k k k k p k p ˆ and ˆ k k k k k p k p ˆ and ˆ
ˆ pˆ (, t) pˆ ˆ (, t)? A) Yes, that is correct! B) No, something is wrong! University of Colorado, Boulder (009)
From the Ehrenfest theorem d P V dt what can you conclude about and Pˆ Ĥ Pˆ A) and commute Pˆ Ĥ Ĥ B) and do not commute C) I cannot make a conclusion University of Colorado, Boulder (009)
A plane wave (I) is normalizable. (II) is a solution of the free-particle Schrödinger equation. (III) represents a state with a definite momentum. Which one is/are true? A) None B) I and II C) I and III D) II and III E) All of them University of Colorado, Boulder (009)
The superposition of two plane waves with k j k (, t 0) (, 0) 1 k t 1/ (, t 0) 1/( ) ep ik j is A) always a periodic function B) is not always a periodic function University of Colorado, Boulder (009)
(, 0) 1/( ) 1/ t ( k )ep Consider If (k) is normalizable, what can you say about the limits of (, t) for A) Goes to zero for all t B) Goes to zero for t=0, but for other t we do not know C) Not enough information to decide. ik dk University of Colorado, Boulder (009)
Consider What is? A) (, 0) 1/( ) 1/ t (, t) (,t) ep[iet / ] () 1/ ( k )ep ik (k)ep ik dk dk B) (,t) C) Something else. 1 (k)ep ik i k t /m dk () 1/ University of Colorado, Boulder (009)
Assume the state of a particle is described by (, 0) 1/( ) 1/ t (k) ( k )ep where is a Gaussian function about k 0 (Δk 0). What can you say about the position and momentum of the particle in this state? A) Both are given with arbitrary precision B) Momentum is given with arbitrary precision, but position has an approimately precise value. C) Position is given with arbitrary precision, but momentum has an approimately precise value. D) Both have approimately precise values. E) Not enough information. ik dk University of Colorado, Boulder (009)
Even without measuring the position; in practice, no instrument can measure the momentum of a particle precisely. There is always an uncertainty in the measurement. Claim: This is due to Heisenberg s uncertainty relation.true or not? A) TRUE B) FALSE University of Colorado, Boulder (009)
Assume the state of a particle is described by where (, 0) 1/( ) 1/ t (k) Does ( k )ep is a Gaussian function about k 0 0 (Δk 0). ( 0, t) A) Yes B) No depend on time? ik dk University of Colorado, Boulder (009)
Assume the state of a particle is described by (, ) 1/( ) 1/ t ( k, t )ep (, t) ( k, t) Compared to, contains A) less information B) more information C) the same information D) cannot be determined / depends ik dk University of Colorado, Boulder (009)
If (, t) ( ) ( t) () is normalized, is normalized too? A) TRUE B) FALSE C) Don t know / depends University of Colorado, Boulder (009)
(, ) (, ) Assume and are stationary states of for different energies E 1 and E.. Consider (, t) 1 t t (, t) 1 (, t) (, t) Is a solution of the time-dependent Schrödinger equation? A) Yes B) No University of Colorado, Boulder (009)
Assume and are stationary states of for different energies E 1 and E.. Consider Is (, t) (, ) (, ) 1 t t (, t) 1 (, t) (, t) a stationary state? A) Yes B) No University of Colorado, Boulder (009)
(, t) ( ) ( t) After assuming a product form solution the time-dependent Schrödinger equation becomes 1 i t m 1 V (, t) If the potential energy function V is a function of time as well as position, i.e. V=V(,t), would separation of variables still work? A) Yes, always B) No, never C) Depends on the functional form of V on and t University of Colorado, Boulder (009)
A general solution of the time-dependent Schrödinger equation can be epanded as (, t) ce E (, t) cee (, t) de E E where are an orthonormal (basis) set of energy eigenfunctions (stationary states). Statement: Because of the superposition principle is a stationary state too? (, t) A) True B) False University of Colorado, Boulder (009)
Assume Ψ 1, Ψ and Ψ 3 are energy eigenstates with E 1, E and E 3 are the corresponding energies. A system is prepared in the following state: Is Ψ normalized? 1 3 1 3 A) Yes B) No University of Colorado, Boulder (009)
Assume Ψ 1, Ψ and Ψ 3 are energy eigenstates with E 1, E and E 3 are the corresponding energies. A system is prepared in the following state: What is the probability to measure E 1? 1 3 1 A) 0 B) 1 3 3 1 3 C) D) E) Cannot be determined 1 University of Colorado, Boulder (009)
Assume Ψ 1, Ψ and Ψ 3 are energy eigenstates with E 1, E and E 3 are the corresponding energies. A system is prepared in the following state: What is the probability to measure E 3? 1 3 1 A) 0 B) 1 3 3 1 3 C) D) E) Cannot be determined 1 University of Colorado, Boulder (009)
The energy eigenstates orthonormal set, i.e. What is * E' ( ) E ( ) d EE ' * () c E E () d? E' E E form an??? A) B) C) E c E c c E' E c E' D) E) ce None of these University of Colorado, Boulder (009)
Assume Ψ 1, Ψ and Ψ 3 are energy eigenstates with E 1, E and E 3 are the corresponding energies. A system is prepared in the following state: 1 3 We do a (first) measurement and measure E. What is the probability to measure E 1 in a subsequent (second) measurement? 1 A) 0 B) 1 3 3 1 3 C) D) E) Cannot be determined 1 University of Colorado, Boulder (009)
What are the basic ideas in QM? Write down some principles you have learned so far and whether they are derived or assumed/postulated! University of Colorado, Boulder (009)
A wave function has been epressed as sum/integral of energy eigenstates: (, t) ce E (, t) cee (, t) de E Compared to the original. the epansion contains: A) More information B) Less information C) The same information D) Cannot be determined/depends University of Colorado, Boulder (009)
A classical particle of energy E approaches a potential energy step of height V 0 ( E <V 0 ). What happens to the particle? V=V 0 V=0 = 0 A) Always moves to the right at constant speed. B) Always moves to the right, but its speed is different for >0 and <0. C) Hits the barrier and reflects D) Hits the barrier and has a chance of reflecting, but might also continue on. E) None of these, or MORE than one of these! E
A 1 ep( ik) A ep( ik) Are and solutions of the following equation? d d ( ) k ( ) 0 A) None of them is a solution B) A 1 ep( ik) is a solution, but A ep( ik) not C) A ep( ) is a solution, but A 1 ep( ik) ik not D) Both are solutions. University of Colorado, Boulder (009)
How was yesterday s eam? A) Too hard B) Hard, but fair C) OK or as epected D) Easy E) Too easy, make the net one harder. University of Colorado, Boulder (009)
How many values for A solve the following equation: A = 1? A) 1 B) C) 4 D) Something else University of Colorado, Boulder (009)
Is A ep( ik) ep( ik) A) Yes B) No C) Don t know an acceptable physical state? University of Colorado, Boulder (009)
What is αδ() (α>0) at =0? A) 0 B) 1 C) α D) E) Something else
For a delta-function potential barrier αδ() at =0: What are the general solutions for (I) <0 and (II) >0? A) (I) A 1 ep(ik) + A ep(-ik) and (II) C 1 ep(ik) + C ep(-ik) B) (I) A 1 ep(ik) + A ep(-ik) and (II) C 1 ep(ik ) + C ep(-ik ) C) (I) A 1 ep(ik) + A ep(-ik) and (II) C 1 ep(κ) + C ep(-κ) D) Something else
For a delta-function potential barrier: What are the correct boundary conditions? A) A 1 + A = C 1 + C and ik (A 1 - A ) = ik (C 1 C ) B) A 1 + A = C 1 + C and ik (A 1 - A ) = ik (C 1 + C ) C) A 1 + A = C 1 - C and ik (A 1 - A ) = ik (C 1 C ) D) A 1 + A = C 1 - C and ik (A 1 - A ) = ik (C 1 + C ) E) None of them
The real part of a stationary state wave function of a particle with energy E is shown below. Assume E > V() for all. Region I Region III Region II In which region is the potential largest? A)Region I B) Region II C) Region III D) Everywhere the same E) Cannot be determined
Consider the case of a potential barrier with the energy of the particle E > V 0. The energy spectrum will be continuous. True (A) or False (B) V() E V 0 0 a
For a particle with E > V 0, incident on a potential barrier, the transmission coefficient is given by: 1 1 T 1 sin 1 with a 4 ( 1) mv 0 / and E / V T is equal to 1 for A) no E B) E only C) many values of E 0
For an infinite square well, the energy spectrum is discrete. What else can you say? (I) The energy levels are equally spaced (II) There is a finite number of energy levels A) Both statements are true B) Both statements are false C) Statement (I) is true, but statement (II) is wrong D) Statement (I) is wrong, but statement (II) is true E) I don t know
The epectation value of the momentum for a particle in the ground state of the infinite square well is. p i A) C) p 0 p k 1 B) D) E) Something else p k 1
Which of the following does represent a ground state wave function of finite square well: A) B) C) -a 0 a -a 0 D) a -a 0 a -a 0 a E) None or more than one of the above
The finite square well has a finite number of energy levels. TRUE (A) or FALSE (B)
The Hamiltonian for the harmonic oscillator is of the following form: ˆ H a ˆ b pˆ This can be rewritten as:, a and b real numbers A) Hˆ aˆ bpˆ a ˆ bpˆ B) Hˆ aˆ ibpˆ a ˆ ibpˆ C) in none of the two forms above
We are about to find the stationary states (and their energies) for the harmonic oscillator. These are the only possible states for a harmonic oscillator (state = normalizable wave function = normalizable solution of the time-dependent Schrödinger equation). TRUE (A) or FALSE (B)
Starting with the ground state and applying successively the raising operator a +, we have found ALL stationary states of the harmonic oscillator. Is this correct? A) Yes, these are all stationary states. B) Well, there might be some other stationary states, which we have missed. C) I am sure, we have missed some stationary states.
Given the following 1-D potential: V V() V 1 -a a Consider the cases: (a) E<V min, (b) V min <E<V 1, (c) V 1 <E<V, (d) E>V Discrete/continuous energy spectrum? Bound or scattering states? General solution in each region? Boundary conditions? Transmission/reflection coefficients? V min
Assume that Ψ 1 and Ψ are solutions of time-dependent Schrödinger equation and α 1 and α are comple numbers. What does the principle of superposition tell you? A) only that Ψ 1 + Ψ is a solution too B) only that α 1 Ψ 1 is a solution too C) α 1 Ψ 1 + α Ψ is a solution too D) None of the above is correct.
If { Φ n >} is a complete basis of a Hilbert space, then Φ n ><Φ n is an operator, TRUE (A) or FALSE(B)? n
A state Ψ> can be represented via its components in a given basis as a vector: a How shall we represent an operator? A) as a scalar B) as a vector C) as a matri D) Something else E) I have no idea a 1 a n
So far, we defined a Hermitian operator via a condition for the epectation value of the corresponding observable. What was the condition? A) A ˆ a B) Aˆ Aˆ * C) Aˆ Aˆ * D) Aˆ A ˆ E) None or more than one of the above are correct
The momentum operator pˆ Hermitian operator, meaning ˆp Is a Hermitian operator? i pˆ is a pˆ (A) Yes (B) No
Consider the Schrödinger equation of a harmonic oscillator : If we change the potential V(), do the eigenvectors of change? A) YES B) NO 1 V() ),, ( ) ( ), ( ), ( m t V t m t t i pˆ
Suppose we have found an energy eigenstate Ψ E (,t) for a given problem. Can this state be written as: E (, t) f ( ) ( p, t) dp where {Φ(p,t)} is a complete set of momentum eigenstates? A) YES B) NO
Consider the following Schrödinger equation: Is it possible to find a set of simultaneous eigenfunctions of the operators and? A) YES B) NO V t V t m t t i ) ( with ), ( ) ( ), ( ), ( Ĥ pˆ
How do the components of the operator for the orbital angular momentum look like? A) B) Lˆ Lˆ p ˆˆ yp ˆˆ y, Lˆ y zp ˆˆ z, yp ˆˆ Lˆ y y, Lˆ z p ˆˆ zˆ pˆ z zp ˆˆ z, Lˆ z p ˆˆ yˆ pˆ y C) D) Lˆ Lˆ yp ˆˆ yp ˆˆ y z zp ˆˆ zp ˆˆ z y,, Lˆ Lˆ y y p ˆˆ zp ˆˆ zp ˆˆ p ˆˆ z z,, Lˆ Lˆ z z p ˆˆ p ˆˆ y yˆ pˆ yˆ pˆ y E) None of the above. 91
Is Lˆ z p ˆˆ y yˆ pˆ a Hermitian operator? A) Yes, and I can show it (using formulas) B) Yes, and I have a good argument. C) No, for sure not. D) Probably not. E) Are you joking? This is a HW problem, but not a clicker question. 91
Is ˆ Lˆ Lˆ Lˆ y z L a Hermitian operator? A) Yes B) No 91
Are the commutators zero or non-zero? (i) :[ yp ˆˆ (ii) z :[ yp ˆˆ, zp ˆˆ z, p ˆˆ A) Both zero B) (i): zero, (ii): non-zero C) (i) : non-zero, (ii): zero D) Both non-zero ] z ] 91
What is the unit of the orbital angular momentum? A) J / s B) J s C) J s / m D) J m / s E) None of the above. 91
For a given λ does the somewhere? Lˆ / Lˆ ladder stop A) Yes, it must stop in both directions B) It will stop in downwards direction, but not in upwards direction C) It will stop in upwards direction, but but not in downwards direction D) In both directions it will not stop 91
Does the following relation hold for a state of the (orbital) angular momentum of a particle: L L y 0 A) Yes, always B) No, never C) Depends on the state
Does the following relation hold for a state of the (orbital) angular momentum of a particle: L z L 0 A) Yes, always B) No, never C) Depends on the state
For a time-independent potential V(r) we have found solutions to the TDSE given by: E with ( r, t) H ( r)ep( iet E E ( r) E ( r) / ) How can the general solution to the TDSE be written (assume a discrete energy spectrum)? A) B) C) ( r, t) ( r, t) ( r)ep( iet / ) D) None of the above E ( r, t) c n E ( r) ep( iet / ) n n ( r, t) c ( r)ep( ie t / ) n E E n E n n 80
For a central potential V(r), H can be written as: 1 H M r r 1 r Mr V ( r) Does H commute with L and/or with L z? A) No, H does not commute with L and not with L z? Lˆ B) H commutes with L but not with L z C) H commutes with L z but not with L D) H commutes with both operators 80