Upper-Division Quantum Mechanics (QM I)

Transformed Course Materials

Learning Goals

If you prefer, you can download these learning goals as a document: [PDF]

Course-Scale Learning Goals

These learning goals were created by a working group of faculty -- both those in physics education research and those in other areas of research. This list represents what we want students to be able to do at the end of the course (as opposed to what content is expected to be covered, as in a syllabus).

1. Math/physics connection: Translate a physical description of a junior-level quantum mechanics problem into the mathematical equation necessary to solve it. Explain the physical meaning of the formal and/or mathematical formulation of and/or solution to a junior-level quantum mechanics problem. Achieve physical insight through the mathematics of a problem.
2. Visualize the problem: Sketch the physical parameters of a problem (e.g., wave function, potential, probability distribution), as appropriate for a particular problem. When presented with a graph of a wave function or probability density, derive appropriate physical parameters of a system.
3. Organized knowledge: Articulate the big ideas from each content area, and/or lecture, thus indicating that they have organized their content knowledge. Filter this knowledge to access the information needed to apply to a particular physical problem. This organizational process should build on knowledge gained in earlier physics classes.
4. Communication: Justify and explain thinking and/or approach to a problem or physical situation, in either written or oral form.
5. Problem-solving techniques: When faced with a quantum mechanics problem, choose and apply appropriate problem solving techniques. Transfer the techniques learned in class and through homework to novel contexts (i.e., to solve problems which do not map directly to those in the book). Justify selected approach (see "Communication" above). In addition to building on techniques learned in previous courses (e.g., recognizing boundary conditions, setting up and solving differential equations, separation of variables, power-series solutions, operator methods), students are expected to develop specific new techniques as listed in concept-scale learning goals below.
1. Approximations: Recognize when approximations are useful, and to use them effectively (e.g., when the energy is very high, or barrier width very wide). Indicate how many terms of a series solution must be retained to obtain a solution of a given order.
2. Symmetries: Recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (e.g., when parity allows you to eliminate certain solutions).
6. Problem-solving strategy: Draw upon knowledge and skills to attack a problem even when a process leading to a correct solution is not yet clear. Continue to develop the ability to monitor progress towards a solution by learning how to:
   • Backtrack and try a new approach when necessary
   • Recognize when a solution has been reached and be able to articulate why this solution is valid (see "Expecting and Checking Solution" below)
   • Persist through to the solution of problems requiring many steps
7. Expecting and checking solution: When appropriate for a given problem, articulate expectations for the solution to a problem, such as:
• The general shape of the wave function
• Dependence on coordinate choice
• Behavior at large distances
• Problem symmetry

For all problems, justify the reasonableness of a solution reached, by using methods such as:
• Checking solution symmetry
• Verifying boundary conditions
• Order of magnitude estimates
• Dimensional analysis
• Limiting or special cases (e.g., checking the solution for correct behavior in limiting or known cases)

8. Intellectual maturity: Students should accept full responsibility for their own learning. They should be aware of what they do and do not understand about physical phenomena and classes of problem. They should learn to ask sophisticated, specific questions. Students should learn to identify and articulate where in a problem they experienced difficulty and to take appropriate action to move beyond that difficulty. Finally, they should regularly check their understanding against these learning goals and seek out appropriate help to fill in any gaps.
9. Coherent Theory: Students should recognize that the material covered in this course sets a framework for a consistent and complete understanding of quantum mechanics.
10. Build on Earlier Material: While the material in the course represents a significant departure from earlier course work both mathematically and conceptually, students should recognize and make use of connections to prior work, techniques and understanding gained in classes in classical physics as well as in their modern physics class.
11. Examples from Recent Research: Whenever possible, examples and homework problems should be drawn from recent research results (nominally defined as the last thirty years).

Topic-Scale Learning Goals

The goals below pertain to specific areas in the study of quantum mechanics which are to be learned in this course. They are organized by subject and thus do not follow any textbook. The subject categories are:

• Mathematics
• Measurement and the quantum state
• The Schrödinger Equation
• Formalism
• Important Systems
• Scattering
• Angular Momentum and Spin

Mathematics

Prerequisites

• Differential Equations:
  • solve straightforward first and second order differential equations using a variety of methods.
  • recognize when separation of variables will simplify a differential equation and correctly apply the technique.
• Complex Numbers: Thoroughly familiar with complex numbers and be able to find the real part, the imaginary part, the complex conjugate and the norm of any complex expression.
• Linear Algebra: Given a matrix operator, find the eigenvalues and eigenvectors of the operator. Not only be able to diagonalize the matrix but be able to explain the physical significance of the procedure and the result.
• Hamiltonian Formalism:
  • Set up the Hamiltonian for a classical system.

Goals

• Statistics: Due to the statistical nature of quantum mechanics, students should be adept at computing probabilities and standard deviations.
• Dirac Delta Function: Correctly compute integrals which contain one or more Dirac delta functions.
• Vector Spaces:
  • Given a set of real or abstract (e.g., Hilbert space) vectors, determine whether the set constitutes a vector space.
  • Given a set of real or abstract (e.g., Hilbert space) vectors, determine whether or not they form a basis of a given vector space.
• Hilbert Space: Compute the correct coefficients of a Hilbert space vector given a basis.
• Operator Theory: Compute the expectation value of an operator in a given state. More generally, compute all the matrix elements of an operator in a given basis. Identify a Hermitian operator.

Measurement and the Quantum State

Goals

• The State Vector:
• Correctly normalize a (normalizable) quantum state.¹
• Describe and calculate different representations of a quantum state (e.g., position space, momentum space).¹
• Observable Operators:
• Know that observable quantities are represented by Hermitian operators.
• Given a wave function and an observable operator, calculate that operator's expectation value.
• For simple systems (e.g., 1-D infinite square well), find the eigenvectors and eigenvalues for the energy operator.
• Measurement Predictions:
• Given the eigenstates of an operator, compute the possible results of a measurement of the observable which corresponds to that operator.¹
• Given a quantum state and the eigenbasis of an observable operator, compute the probabilities of obtaining the possible values which would result from a measurement of the corresponding observable quantity.
• Given the results of a repeated measurement of an observable on a quantum state, construct a plausible quantum state as a superposition of the eigenstates of the operator associated with the observable.¹
• Measurement Effects:
• Describe what is known about the state of a system immediately after a measurement, including the significance of the measured value.¹
• Time Evolution:
• Given an initial wave function and a basis of energy eigenstates, find the time-dependent wave function.
• Given an initial wave function and a basis of energy eigenstates, deduce when the probability distribution of an operator will be time dependent.¹
• Operator Commutation and Compatibility:
• Describe the relationship which must exist between two operators in order for a common eigenbasis to exist.
• Compute the commutator of the position and momentum operators as well as the commutation relationships between angular momentum operators.
• Describe the effect of following the measurement of an observable with the measurement of an incompatible operator.¹
• Given two non-commuting observables, A and B and the result of a measurement of A, compute the possible outcomes of a subsequent measurement of B along with the appropriate probabilities.

Schrödinger Equation

Goals

• Time Dependent Schrödinger Equation: Use the time dependent Schrödinger Equation to compute the time evolution of a wave function.
• Time Independent Schrödinger Equation: Describe the conditions under which separation of variables can be used to create a time independent Schrödinger Equation and use this equation to:
  • solve for the energy levels of the system
  • apply boundary conditions and solve for the stationary states (energy eigenstates) of the system
  • apply the Hamiltonian and boundary conditions to determine whether the energy eigenstates are discrete or continuous.
  • specify the evolution in time of a system when both an initial state and the energy eigenstates known

Formalism

Goals

• Normalization:
  • Explain the relationship between the normalization of a wave function and the ability to correctly calculate expectation values or probability densities.
  • Correctly normalize any wave function which represents a physically realizable state.
• Hamiltonian:
  • Set up the Hamiltonian for a quantum mechanical system when they can calculate the potential energy for the corresponding classical system.¹
  • Use commutation relations to be able to determine which operators have eigenstates which are time independent.
• Uncertainty Principle:
  • Given a quantum state and an observable, compute the uncertainty (standard deviation in the measurement) of the observable.
  • Given two observables, compute the minimum uncertainty of measuring both observables on any quantum state.
• Probability in Quantum Mechanics:
  • Given a (time-dependent) wave function, compute the time-dependent probability density.
  • For a given quantum state, compute the probability of measuring any particular value for any common observable.¹

Important Systems

Goals

• Infinite Square Well: Thoroughly familiar with all aspects of the one dimensional infinite square well.
• Given the size and position of the potential, compute the energy eigenvalues and the energy eigenstate position-space wave functions.
• Compute the time evolution of a superposition of energy eigenstates as well as the expectation value of common observables for a superposition state.
• General One-dimensional Systems:
  • Given a one-dimensional potential, sketch the first few energy eigenstates.
• Harmonic Oscillator:
  • Given a specific harmonic-oscillator potential, compute the energy eigenvalues.
  • Given the raising and lowering operators, find the lowest energy eigenstate.
  • Given the raising and lowering operators and an energy-eigenstate wave function, find the energy eigenstates on either side.
  • Sketch the first few energy eigenstates of the harmonic oscillator.
  • Compute position and momentum expectation values using the raising and lowering operators.
• Free Particle: Adept at using the position-space and momentum-space wave functions of the free particle. In particular, use them to construct wave packets.
• Two-State Systems:
  • Given a two-dimensional Hamiltonian, find its eigenstates and eigenvalues.
  • Given a two-state system in a superposition state, correctly compute the probabilities of measuring each eigenvalue.
• Hydrogen Atom:
  • Set up the Schrödinger equation for a hydrogen-like atoms.
  • Perform variable separation on the SE for hydrogen.
  • Describe the energy eigenstates for hydrogen-like atoms including the significance and use of their quantum numbers.

Angular Momentum and Spin

Goals

• Angular Momentum in Quantum Mechanics:
  • Compute the angular momentum of a system in a known eigenstate of an angular momentum operator (e.g., L², L_z)
  • Given a system in a known state, compute the probabilities of the possible results of measuring an angular momentum observable (e.g., L², L_z, L_y)
• Spin:
  • Given a system in a known state, compute the probabilities of the possible results of measuring a spin observable (e.g., S², S_z, S_y

 

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¹Targeted in QMAT