## 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*) = (*Aψ*+_{1}*Bψ*)_{2}^{(-iEt/ℏ)}when ψ_{1}and ψ_{2}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 S
_{x}” or “Eigenstates of S_{x}are orthogonal to eigenstates of S_{y}”.^{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

- Singh, C., Student understanding of quantum mechanics. Am. J. Phys., 2001.
**69**(8): p. 885-896. - 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.
- Goldhaber, S., et al.,
*Transforming Upper-Division Quantum Mechanics: Learning Goals and Assessment.*2009 PERC Proc, 2009.**1179**: p. 145-148. - Crouse, A.D., Research on Student Understanding of Quantum Mechanics as a Guide for Improving Instruction. 2007, University of Washington.
- 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.
- Gire, E. and C. Manogue, Resources Students Use to Understand Quantum Mechanical Operators. 2008.
- 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. - 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.