Electrostatics

Challenges to Student Success

This page provides a brief summary of common student learning difficulties which have been previously published in studies of student learning in upper-division quantum mechanics. Because many of the documented difficulties fall into well defined categories such as time development, most of the difficulties are grouped into subject-specific tables.

If you prefer, you can download the full list of student difficulties as a document: [DOC] [PDF].

Time Development

  • If the system is initially in an eigenstate of an operator Ĝ, then the expectation value of that operator is time independent.1
  • An eigenstate of any operator is a stationary state.1
  • If the system is in an eigenstate of any operator Ĝ, then it remains in the eigenstate of Ĝ forever unless an external perturbation is applied.1
  • If the time dependent exponential factors cancel out in the expectation value for an operator Ĝ, then the system (e.g., the wave function) does not evolve. 1
  • All allowable quantum states have a definite energy, even states which are superposition of energy eigenstates with different energies. An example of this type of statement is Ψ(x,t) = (Aψ1 + Bψ2)(-iEt/ℏ) when ψ1 and ψ2 are energy eigenstates with different energies.2
  • There is no relationship between the Hamiltonian operator and the time development of a quantum state.3
  • There is no time dependence in the position probability density of a superposition state.2
  • Even after a measurement, an undisturbed quantum system will eventually return to its initial state.4
  • Any undisturbed wave function will spread out, eventually producing a flat probability density (either zero or non-zero).4
  • An undisturbed quantum system will eventually drop into a stationary state.4
  • When a system is in an energy eigenstate, the expectation value of an operator may depend upon time.1
  • If the expectation value of an operator Ĝ is zero in some initial state, the expectation value cannot have any time dependence.5
  • The time evolution of an arbitrary state cannot change the probability of obtaining a particular outcome when any observable is measured.1
  • If the system is initially in an eigenstate of any operator Ĝ, then the expectation value of another operator Ĵ will be time independent if Ĝ and Ĵ commute.1

Measurement

  • When an operator acts on a state it is a mathematical representation of making a measurement of the physical quantity associated with the operator.6
  • After any measurement, all observables have definite values.4
  • It is not possible to determine the probability distribution of possible measurements of an operator, Ĵ, given a quantum state expressed in the basis of operator Ĝ if Ĝ and Ĵ do not commute.4

Wave Functions, Potentials, and Probability

  • The potential energy graph of a region of space is the same regardless of the particle (e.g., electron, positron, neutron) placed in that region.7
  • Local deBroglie wavelength does not vary with kinetic energy.5
  • A particle will “spend more time” in a region of lower potential energy.5
  • A system will have discrete energy levels even when there are too few boundary conditions.7
  • Certain special functions (e.g., the energy eigenstates of the infinite square well) are stationary states for all potentials.4

Other Learning Difficulties

  • Every quantum state has a well-defined energy.1
  • A quantum system with two classical turning points admits bound states (much like a classical system with the same potential), regardless of that potential’s behavior at infinity.2,7,8
  • All operators have the same eigenfunctions.4
  • An object with a label “x” is orthogonal to or cannot influence an object with a label “y”. Examples of this type of statement are student responses such as “The magnetic field is in the z direction so an electron is not influenced if it is initially in an eigenstate of Sx” or “Eigenstates of Sx are orthogonal to eigenstates of Sy”.2
  • Successive measurements of continuous variables of a particle produce 'somewhat' deterministic outcomes (e.g., a quickly repeated position measurement will yield a value close to the original) whereas successive measurements of discrete variables, can produce very different outcomes (e.g., a quickly repeated measurement of a particle's spin in the z direction could yield a value opposite from the original measurement).2

References

  1. Singh, C., Student understanding of quantum mechanics. Am. J. Phys., 2001. 69(8): p. 885-896.
  2. Singh, C., Transfer of Learning in Quantum Mechanics, in 2004 PERC Proc, J. Marx, P. Heron, and S. Franklin, Editors. 2005, AIP: Sacramento, California (USA). p. 23-26.
  3. Goldhaber, S., et al., Transforming Upper-Division Quantum Mechanics: Learning Goals and Assessment. 2009 PERC Proc, 2009. 1179: p. 145-148.
  4. Crouse, A.D., Research on Student Understanding of Quantum Mechanics as a Guide for Improving Instruction. 2007, University of Washington.
  5. Singh, C., Assessing and improving student understanding of quantum mechanics, in 2005 PERC Proc, P. Heron, L. McCullough, and J. Marx, Editors. 2006, AIP. p. 69-72.
  6. Gire, E. and C. Manogue, Resources Students Use to Understand Quantum Mechanical Operators. 2008.
  7. McKagan, S.B. and C.E. Wieman. Exploring Student Understanding of Energy through the Quantum Mechanics Conceptual Survey. in 2005 PERC Proc. 2006. Melville, NY: AIP Press.
  8. Cataloglu, E. and R.W. Robinett, Testing the development of student conceptual and visualization understanding in quantum mechanics through the undergraduate career. Am. J. Phys., 2002. 70(3): p. 238-251.