Generalized Uncertainty: From Cauchy-Schwarz to Observables

Introduction

In our last lesson, we explored the profound consequences of quantum measurement, focusing on the Born rule and the projection postulate. We established that sequential measurements of incompatible, or non-commuting, observables are order-dependent and that the act of measuring one can fundamentally alter the system, erasing information about the other. This "measurement disturbance" is not a flaw in our instruments but a fundamental feature of the quantum world.

Today, we will quantify this inherent trade-off. We will derive the generalized uncertainty relation, a cornerstone of quantum mechanics that provides a precise mathematical statement about the limits of our simultaneous knowledge of any two observables. You will see that this principle is not an additional axiom but a direct and elegant consequence of the Hilbert space structure we've been developing, derived using the powerful Cauchy-Schwarz inequality.

Defining Uncertainty

Before we can derive a principle about uncertainty, we must first agree on a precise, quantitative definition. In statistics, the "spread" or "uncertainty" in a set of measurements of a random variable is quantified by its standard deviation. We adopt the same definition in quantum mechanics.

For an observable represented by the Hermitian operator $A^$ , its uncertainty in a given state $∣ ψ ⟩$ is defined as the root mean square (RMS) deviation from its expectation value, $⟨ A^⟩ = ⟨ ψ ∣ A^∣ ψ ⟩$ . The variance, $(Δ A)^{2}$ , is given by:

(Δ A)^{2} = ⟨(A^- ⟨ A^⟩)^{2} ⟩ = ⟨ ψ ∣ (A^- ⟨ A^⟩)^{2} ∣ ψ ⟩

The uncertainty, $Δ A$ , is simply the square root of the variance.

To simplify our upcoming derivation, it's convenient to define a "deviation operator," $a^= A^- ⟨ A^⟩$ . Since $A^$ is Hermitian and $⟨ A^⟩$ is a real number, the operator $a^$ is also Hermitian. In this more compact notation, the variance becomes $(Δ A)^{2} = ⟨ a^^{2} ⟩$ .

The following resource provides a clear statement of this definition.

3.2: General Uncertainty Principal

These notes by Michael Fowler lay out the standard definition of uncertainty that we will use as the starting point for our derivation.

Please read the brief introductory section, "A Quantitative Measure of “Uncertainty”. Focus on how the variance (\Delta A)^2 is defined and how the deviation operator \hat{a} simplifies its expression.

The Generalized Uncertainty Principle: Derivation

With our definitions in place, we are now ready to derive the generalized uncertainty relation for two arbitrary Hermitian operators, $A^$ and $B^$ . The proof is a beautiful application of the Cauchy-Schwarz inequality to the abstract vector space of quantum states.

The overall strategy is as follows:

Define two new state vectors based on the action of the deviation operators $a^$ and $b^$ on the system's state $∣ ψ ⟩$ .
Express the product of the variances, $(Δ A)^{2} (Δ B)^{2}$ , as a product of the squared norms of these new vectors.
Apply the Cauchy-Schwarz inequality to this product.
Decompose the resulting inner product into its real and imaginary parts, which relate directly to the anti-commutator and commutator of $A^$ and $B^$ .
Use the fact that the squared magnitude of a complex number is always greater than or equal to the squared magnitude of its imaginary part to arrive at the final inequality.

The following video provides an excellent, step-by-step walkthrough of this exact derivation. It meticulously explains each step and the properties of the operators involved.

The Generalized Uncertainty Principle | Proof/Derivation

This video, "The Generalized Uncertainty Principle | Proof/Derivation" from the Faculty of Khan channel, will guide you through the entire proof. It's very systematic and aligns perfectly with the rigorous approach you appreciate.

Please watch the video from the beginning until the final result. Pay close attention to these key stages: Setup and Definitions: This part introduces the deviation operators and their relation to the variance, as we discussed. It also shows how the product of deviation operators can be split using the commutator and anti-commutator. Applying Cauchy-Schwarz: The core of the proof begins at this point. Follow carefully how the abstract vectors |F\rangle and |G\rangle are defined and how the Cauchy-Schwarz inequality is applied to their inner products. Final Steps: The derivation is completed by linking the result from the Cauchy-Schwarz inequality to the commutator. This involves recognizing the expectation value of the commutator as the imaginary part of a complex number.

For your reference, the same proof is written out concisely in the resource by Fowler. You can use it to solidify your understanding of the steps shown in the video.

3.2: General Uncertainty Principal

This text provides a compact, written version of the derivation you just watched. Reading through it will help reinforce the logic.

Now, please read the section titled "The Generalized Uncertainty Principle", which contains the full proof. Trace how each line corresponds to the steps outlined in the video.

Let's summarize the result. For any two Hermitian operators $A^$ and $B^$ , and for any state $∣ ψ ⟩$ , the following inequality holds:

(Δ A)^{2} (Δ B)^{2} \geq \frac{1}{2 i} ⟨[A^, B^]⟩^{2} = \frac{1}{4} ∣ ⟨[A^, B^]⟩ ∣^{2}

Taking the square root gives the more common form:

Δ A Δ B \geq \frac{1}{2} ∣ ⟨[A^, B^]⟩ ∣

This powerful result is known as the Robertson-Schrödinger uncertainty relation. It formalizes the intuition we built in the last lesson: the degree to which we are fundamentally limited in our simultaneous knowledge of two observables is determined by their commutator. If $[A^, B^] = 0$ , the lower bound is zero, and states can exist where both $Δ A$ and $Δ B$ are zero. If they do not commute, no such state is possible.

The Heisenberg Uncertainty Principle

The most famous application of the generalized uncertainty principle involves the position ( $x^$ ) and momentum ( $p^_{x}$ ) operators. As you know from your prior studies, their commutator is a fundamental constant of nature:

[x^, p^_{x}] = i ℏ

Substituting this into our general formula gives:

Δ x Δ p_{x} \geq \frac{1}{2} ∣ ⟨ i ℏ ⟩ ∣ = \frac{1}{2} ∣ i ℏ∣ = \frac{ℏ}{2}

This is the celebrated Heisenberg uncertainty principle. It is not a separate postulate, but a specific instance of the more general relation we just derived. It asserts that it is impossible to prepare a quantum state in which a particle has both a definite position and a definite momentum. The more precisely one is known, the less precisely the other can be specified.

Minimum Uncertainty States

A natural question arises: Are there states that live on the boundary of this inequality, where $Δ x Δ p_{x} = ℏ/2$ ? These are known as minimum uncertainty states.

For the inequality to become an equality, both of the inequalities used in the derivation must themselves become equalities. Tracing the proof back, this requires two conditions to be met simultaneously:

The Cauchy-Schwarz inequality must be saturated. This occurs if the vectors $∣ ψ_{a} ⟩ = a^∣ ψ ⟩$ and $∣ ψ_{b} ⟩ = b^∣ ψ ⟩$ are linearly dependent, i.e., $∣ ψ_{b} ⟩ = λ ∣ ψ_{a} ⟩$ for some complex scalar $λ$ .
The inequality $∣ z ∣^{2} \geq ∣ Im (z) ∣^{2}$ must be saturated for $z = ⟨ ψ_{a} ∣ ψ_{b} ⟩$ . This requires the real part of $⟨ ψ_{a} ∣ ψ_{b} ⟩$ to be zero.

As shown in Fowler's notes, satisfying these two conditions for the position and momentum operators leads to a first-order differential equation for the wave function $ψ (x)$ . The solution to this equation is a Gaussian wave packet.

3.2: General Uncertainty Principal

This final section of the notes investigates the conditions under which the uncertainty principle becomes an equality.

Please read the section titled "Minimizing the Uncertainty". Follow the argument that shows how the two conditions for equality lead to a differential equation whose solution is a Gaussian wave function.

This is a profound result. The Gaussian wave packet, a function of central importance in many areas of physics and mathematics, emerges here as the unique quantum state that provides the optimal trade-off between position and momentum localization. You will encounter this concept again when studying coherent states of the quantum harmonic oscillator, which are also minimum uncertainty states that follow classical trajectories.

Conclusion

In this lesson, we have formalized one of the most distinctive and foundational principles of quantum mechanics. By applying the Cauchy-Schwarz inequality to the operator formalism, we derived the generalized uncertainty relation from first principles.

Key Takeaways:

The uncertainty of an observable $A^$ is defined as its standard deviation in a given state: $Δ A = ⟨(A^- ⟨ A^⟩)^{2} ⟩$ .
The generalized uncertainty principle, $Δ A Δ B \geq \frac{1}{2} ∣ ⟨[A^, B^]⟩ ∣$ , provides a fundamental lower bound on the product of uncertainties for any two observables.
This relation confirms that the commutator, $[A^, B^]$ , is the mathematical embodiment of the compatibility of observables.
The Heisenberg uncertainty principle, $Δ x Δ p_{x} \geq ℏ/2$ , is a direct consequence of the canonical commutation relation $[x^, p^_{x}] = i ℏ$ .
For position and momentum, the states that saturate this inequality—minimum uncertainty states—are Gaussian wave packets.

Having established the core tenets of quantum states and measurements, our next step is to explore their evolution in time. In the upcoming lesson, we will derive the time-evolution operator from the Schrödinger equation and establish its unitarity, a crucial property that ensures the conservation of probability over time.

Recommended resources

Introduction

Defining Uncertainty

The Generalized Uncertainty Principle: Derivation

The Heisenberg Uncertainty Principle

Minimum Uncertainty States

Conclusion