Chemistry 352

Postulate 1: Wavefunctions

According to Postulate 1, the physical state of a quantum system is described by a wavefunction \( \psi(x,t) \). However, not every mathematical function is an acceptable wavefunction. Only functions with certain properties can represent physical states.

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Properties of an acceptable wavefunction

An acceptable wavefunction must satisfy the following conditions:

• Single-valued: The wavefunction must have a single, unique value at every point in space. This ensures that the probability density \( |\psi(x,t)|^2 \) is well defined.
• Finite: The wavefunction must remain finite everywhere. An infinite value of \( \psi \) would correspond to an infinite probability density, which is unphysical.
• Continuous: The wavefunction must be continuous in space. Sudden jumps in \( \psi \) would imply undefined physical behavior.
• Smooth (piecewise differentiable): The first derivative of the wavefunction must be continuous wherever the potential energy is finite. This condition ensures that the kinetic energy operator can act on the wavefunction.

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Normalizability

The wavefunction must also be normalizable. This means that the total probability of finding the particle somewhere in space must be equal to one.

\[ \int |\psi(x,t)|^2\,dx = 1. \]

If a wavefunction does not initially satisfy this condition, it can be multiplied by a constant so that it does. This process is called normalization However, it is important to not that in order to be normalizable, the square of a wavefuntion \(\lvert\psi^* \psi\rvert\) must vanaish at the ends of the relevant space. If it does not, the squared function will have an infinite area under the curve, making it impossible to notmalize!

A function that cannot be normalized (because the integral of \( |\psi|^2 \) diverges) cannot represent a physical state. Such a function would predict an infinite total probability, which is not meaningful.

Big idea: acceptable wavefunctions are mathematical functions with physical constraints, and normalizability ensures a consistent probabilistic interpretation.

Postulate 2: Operators

According to Postulate 2, every measurable physical quantity (observable) is represented by an operator. Operators act on wavefunctions to extract physical information, such as position, momentum, and energy.

Not all operators are suitable for representing observables. In quantum mechanics, observable operators must satisfy two important mathematical properties: they must be linear and Hermitian.

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Linear operators

An operator \( \hat{A} \) is said to be linear if it satisfies the superposition property:

\[ \hat{A}\left(c_1\psi_1 + c_2\psi_2\right) = c_1\,\hat{A}\psi_1 + c_2\,\hat{A}\psi_2, \]

where \(c_1\) and \(c_2\) are constants. Linearity ensures that if two wavefunctions are allowed states, then any linear combination of them is also an allowed state. This property underlies the principle of superposition in quantum mechanics.

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Hermitian operators

An operator \( \hat{A} \) is Hermitian if it satisfies

\[ \int \psi^*(x)\,\hat{A}\phi(x)\,dx = \int \left(\hat{A}\psi(x)\right)^* \phi(x)\,dx \]

for all acceptable wavefunctions \( \psi \) and \( \phi \), assuming appropriate boundary conditions. Hermitian operators are the quantum-mechanical analog of real-valued observables in classical physics.

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Consequences of Hermiticity

Hermitian operators have several crucial physical consequences:

• Real eigenvalues: The eigenvalues of a Hermitian operator are always real. This ensures that measured values of physical observables are real numbers.
• Orthogonal eigenfunctions: Eigenfunctions corresponding to different eigenvalues are orthogonal: \( \int \psi_m^*\psi_n\,dx = 0 \) for \( m \neq n \).
• Complete set of states: The eigenfunctions of a Hermitian operator form a complete set, meaning any acceptable wavefunction can be written as a linear combination of them.

These properties allow quantum states to be expanded in terms of eigenfunctions of observables and make it possible to predict measurement outcomes and probabilities.

Big idea: observables are represented by linear, Hermitian operators, ensuring real measurement outcomes and a well-defined mathematical structure for quantum states.

Postulate 3: Observable Properties

Postulate 3 describes how measurements are made in quantum mechanics and how measurement outcomes relate to the mathematical structure of operators and wavefunctions.

When an observable corresponding to an operator \( \hat{A} \) is measured, the result is not arbitrary. Instead, the measurement outcomes are determined by the eigenvalue equation

\[ \hat{A}\psi_n = a_n\,\psi_n, \]

where \( a_n \) are the eigenvalues and \( \psi_n \) are the corresponding eigenfunctions of \( \hat{A} \).

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Measurement outcomes

A single measurement of the observable \(A\) can yield only one of the eigenvalues \( a_n \) of the operator \( \hat{A} \). No other values are possible.

If the system is already in an eigenstate \( \psi_n \) of the operator being measured, the measurement outcome is certain and will always return the eigenvalue \( a_n \).

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Measurement probabilities

If the system is in a general state \( \Psi \) that is not an eigenfunction of \( \hat{A} \), the wavefunction can be expanded in terms of the eigenfunctions of \( \hat{A} \):

\[ \Psi = \sum_n c_n\,\psi_n. \]

The probability of obtaining the eigenvalue \( a_n \) in a measurement is \( |c_n|^2 \), where

\[ c_n = \int \psi_n^*(x)\,\Psi(x)\,dx. \]

These probabilities reflect the inherently probabilistic nature of quantum measurements.

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Wavefunction collapse

Immediately after a measurement yielding the value \( a_n \), the wavefunction collapses into the corresponding eigenfunction \( \psi_n \).

If the same observable is measured again immediately after, the result will be the same eigenvalue with certainty. However, measuring a different, non-commuting observable can change the state.

Big idea: measurement outcomes are restricted to eigenvalues, probabilities are determined by overlaps with eigenstates, and measurement fundamentally changes the quantum state.

Postulate 4: Expectation Values

Postulate 4 provides the rule for calculating the average value of a physical observable when a system is described by a wavefunction \( \psi(x,t) \). This average is called the expectation value.

Expectation values do not generally correspond to the result of a single measurement. Instead, they represent the average outcome obtained from many identical measurements performed on systems prepared in the same state.

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Definition of the expectation value

For an observable \(A\) represented by the operator \( \hat{A} \), the expectation value is defined as

\[ \langle A \rangle = \int \psi^*(x,t)\,\hat{A}\,\psi(x,t)\,dx. \]

This expression combines the wavefunction, the operator corresponding to the observable, and complex conjugation in a way that ensures the result is real for Hermitian operators.

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Connection to probability

If the operator corresponds to position (\( \hat{A}=x \)), the expectation value reduces to

\[ \langle x \rangle = \int x\,|\psi(x,t)|^2\,dx, \]

which is the familiar average of a probability distribution. In this case, the expectation value is the mean position of the particle.

For more general observables, the operator modifies the wavefunction before the probability weighting is applied.

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Expectation values and eigenstates

If the system is in an eigenstate \( \psi_n \) of the operator \( \hat{A} \), then the expectation value is equal to the corresponding eigenvalue:

\[ \langle A \rangle = a_n. \]

In this case, repeated measurements of the observable always give the same result, and there is no uncertainty in the measurement.

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Expectation values and uncertainty

Expectation values are also used to quantify the spread in measurement outcomes. The uncertainty (standard deviation) in an observable \(A\) is defined as

\[ (\Delta A)^2 = \langle A^2 \rangle - \langle A \rangle^2. \]

This expression measures how widely the possible measurement outcomes are distributed around the expectation value.

Big idea: expectation values connect the mathematical wavefunction to measurable averages and provide the foundation for understanding uncertainty in quantum mechanics.

Postulate 5: Time Dependence

Postulate 5 describes how a quantum system evolves in time. While the previous postulates explain how states are represented and how measurements are made, this postulate provides the fundamental dynamical law of quantum mechanics.

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The time-dependent Schrödinger equation

The time evolution of a quantum state \( \psi(x,t) \) is governed by the time-dependent Schrödinger equation:

\[ i\hbar\,\frac{\partial \psi(x,t)}{\partial t} = \hat{H}\psi(x,t), \]

where \( \hat{H} \) is the Hamiltonian operator, representing the total energy (kinetic plus potential) of the system.

Given an initial wavefunction \( \psi(x,0) \), the Schrödinger equation uniquely determines the wavefunction at all later times.

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Time-independent Schrödinger equation

If the potential energy does not depend on time, the Hamiltonian is time independent, and solutions can be separated into spatial and time parts. In this case, the wavefunction can be written as

\[ \psi(x,t) = \psi_n(x)\,e^{-iE_n t/\hbar}. \]

Substituting this form into the time-dependent Schrödinger equation leads to the time-independent Schrödinger equation:

\[ \hat{H}\psi_n(x) = E_n\,\psi_n(x). \]

The solutions \( \psi_n(x) \) are called stationary states, and the corresponding energies \( E_n \) are the allowed energy levels of the system.

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Physical meaning of stationary states

In a stationary state, the probability density \( |\psi(x,t)|^2 \) is independent of time, even though the wavefunction itself depends on time through a complex phase factor.

As a result, all expectation values of observables that do not explicitly depend on time are constant in time for stationary states.

If a system is prepared in a superposition of stationary states, the probability density generally becomes time dependent, leading to observable motion and oscillatory behavior.

Big idea: the Schrödinger equation governs the time evolution of quantum states, and stationary states provide stable, time-independent probability distributions.

Key points (one glance)

• The state of a quantum system is fully described by a wavefunction \( \psi(x,t) \); the probability density is \( |\psi(x,t)|^2 \).
• Acceptable wavefunctions must be single-valued, finite, continuous, and normalizable.
• Every observable is represented by a linear, Hermitian operator with real eigenvalues and orthogonal eigenfunctions.
• A measurement of an observable can yield only one of the operator’s eigenvalues, with probabilities determined by overlaps with the corresponding eigenstates.
• After a measurement, the wavefunction collapses into the eigenstate associated with the measured value.
• Expectation values give the average result of many measurements and are calculated as \( \langle A \rangle = \int \psi^* \hat{A} \psi\,dx \).
• Uncertainty in an observable is quantified by \( (\Delta A)^2 = \langle A^2\rangle - \langle A\rangle^2 \).
• The time evolution of a quantum state is governed by the time-dependent Schrödinger equation.
• For time-independent Hamiltonians, stationary states have time-independent probability densities.
• Big picture: the postulates define how quantum states are described, measured, and evolved in time.