PHYS 3220 - Quantum I
Student Learning Difficulties
This document 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. In these tables, the
phrase, “some students say that …”, may also be read as “answers provided by some students are
consistent with the believe that …”.
Student learning difficulties related to time development in quantum mechanics
Student Learning Difficulties: Time Development
Students believe that if the system is initially in an eigenstate of an operator Ĝ, then the
expectation value of that operator is time independent.1
Some students say that an eigenstate of any operator is a stationary state.1
Some students say that 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
Some students say that 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
Some students say that 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 (x,t)  (A1  B2 )exp(iEt / )when 1 and 2 are energy eigenstates with
different energies.2
Some students say that there is no relationship between the Hamiltonian operator and the time
development of a quantum state.3
 students say that there is no time dependence in the position probability density of a
Some
superposition state.2
Some students say that even after a measurement, an undisturbed quantum system will
eventually return to its initial state.4
Some students say that any undisturbed wave function will spread out, eventually producing a
flat probability density (either zero or non-zero) .4
Some students say that an undisturbed quantum system will eventually drop into a stationary
state.4
Some students say that when a system is in an energy eigenstate, the expectation value of an
operator may depend upon time.1
Some students say that if the expectation value of an operator Ĝ is zero in some initial state,
the expectation value cannot have any time dependence.5
Some students say that the time evolution of an arbitrary state cannot change the probability of
obtaining a particular outcome when any observable is measured.1
Some students say that 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
Student learning difficulties related to measurement in quantum mechanics
Student Learning Difficulties: Measurement
Some students say that 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
Some students say that after any measurement, all observables have definite values.4
Some students say that 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
Student learning difficulties related to wave functions, potentials, and probability in quantum
mechanics
Student Learning Difficulties: Wave Functions, Potentials, and Probability
Some students say that 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
Some students say that local deBroglie wavelength does not vary with kinetic energy.5
Some students say that a particle will “spend more time” in a region of lower potential
energy.5
Some students say that a system will have discrete energy levels even when there are too few
boundary conditions.7
Some students say that certain special functions (e.g., the energy eigenstates of the infinite
square well) are stationary states for all potentials.4
Other student learning difficulties in upper-division quantum mechanics
Other Student Learning Difficulties
Some students say that every quantum state has a well-defined energy.1
Some students say that 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
Some students say that all operators have the same eigenfunctions.4
Some students say that 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
Some students say that 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. 885896.
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.