9- CHEM-327 Lecture 9: Quantum Operators, Measurement, and Expectation Values

Review Question

• The wave function: Is an eigenfunction for:

  • a) momentum and kinetic energy

  • b) momentum but not kinetic energy

  • c) kinetic energy but not momentum

  • d) neither momentum nor kinetic energy

  • (The answer to this question will be implicitly covered by the following discussion on eigenfunctions and operators.)

Standing Wave

• Consider a graph of $\psi(x)$:

  • The wave function has become real-valued.

  • An interference pattern arises from the combination of a forward and a backward traveling wave.

    • These waves possess both positive and negative momentum.

  • The resulting wave created by this combination is known as a standing wave.

Hermitian Operators

• Definition: Hermitian operators are operators that satisfy the condition: $\int \phi^* A \psi \text{d}x = \int (A \phi)^* \psi \text{d}x$

  • Where $A$ is the operator, $\psi$ and $\phi$ are wave functions, and $*$ denotes the complex conjugate.

• Key Property: It can be rigorously shown that Hermitian operators possess real numbers as Eigenvalues.

• Physical Significance: Since all physical quantities that can be measured are real-valued, the quantum mechanical operators that represent these physical quantities must be Hermitian.

• Example: The text mentions that an example demonstrating the momentum operator ($\hat{p}$) as Hermitian can be found in the textbook.

Orthogonal Wave Functions

• Definition: Two wave functions, $\phi<em>n$ and $\phi</em>m$, are considered orthogonal if their inner product is zero: $\int \phi<em>n^* \phi</em>m \text{d}x = 0$

  • If $n = m$, and the wave function is normalized, then $\int \phi<em>n^* \phi</em>n \text{d}x = 1$. If it is not normalized, it will be some non-zero constant.

• Connection to Hermitian Operators: It can be demonstrated that two eigenfunctions belonging to different eigenvalues of a Hermitian operator are always orthogonal.

Measuring Values (I) - Eigenfunctions

• Scenario: If a wave function $\psi$ is an eigenfunction of an operator $A$, meaning $A \psi = a \psi$ where $a$ is the eigenvalue.

• Measurement Outcome: In this specific case, the quantity corresponding to operator $A$ will be measured precisely by determining its Eigenvalue $a$.

  • There is no uncertainty in the measurement result.

Measuring Values (II) - Superposition

• Scenario: Often, the starting wave function $\psi$ is NOT an eigenfunction of a particular operator $A$.

• Superposition Principle: However, any valid wave function $\psi$ can always be expressed as a superposition (linear combination) of the eigenfunctions ($\phi<em>n$) of that operator $A$: $\psi = \sum</em>n c<em>n \phi</em>n$

  • Here, $c_n$ are complex coefficients, known as probability amplitudes.

  • The $\phi<em>n$ are the eigenfunctions of operator $A$, corresponding to eigenvalues $a</em>n$.

• Measurement Uncertainty: If the eigenfunctions ($\phi<em>n$) in the superposition correspond to different eigenvalues ($a</em>n$), then the value for the quantity represented by operator $A$ cannot be measured exactly in a single measurement.

  • Instead, upon measurement, the system will collapse into one of the eigenstate $\phi<em>n$, and the measured value will be the corresponding eigenvalue $a</em>n$.

Probability for a Measured Value

• Calculation: When $\psi$ is expressed as a superposition of normalized eigenfunctions ($\phi<em>n$) of operator $A$ (i.e., $\psi = \sum</em>n c<em>n \phi</em>n$),

• Probability Formula: The probability of measuring a specific eigenvalue $a<em>n$ (corresponding to eigenfunction $\phi</em>n$) is given by: $P(a<em>n) = |c</em>n|^2$

  • This formula holds when both $\psi$ and the $\phi_n$ are normalized wave functions.

• Task/Projection: Using the orthogonality of the eigenfunctions $\phi<em>n$, the coefficient $c</em>n$ can be found by calculating the following integral: $c<em>n = \int \phi</em>n^* \psi \text{d}x$

  • This calculation is often referred to as calculating the "projection" of $\psi$ onto $\phi_n$.

  • It determines the probability amplitude $c<em>n$ and, thus, how much of the eigenfunction $\phi</em>n$ is contained within the initial wave function $\psi$.

Expectation Value

• Definition: The expectation value ($\langle A \rangle$) represents the average expected value for a measurement of the quantity associated with operator $A$ over many identical systems or measurements.

• Formula (using coefficients): It can be determined by: $\langle A \rangle = \sum<em>n |c</em>n|^2 a_n$

  • Where $|c<em>n|^2$ is the probability of measuring eigenvalue $a</em>n$.

• Equivalent Formula (using operators): This is equivalent to (proof not discussed): $\langle A \rangle = \int \psi^* A \psi \text{d}x$

  • This formula requires the wave function $\psi$ to be normalized.