Expectation Value In Quantum Mechanics $⟨Ψ|A|Ψ⟩$ Explained

At the heart of quantum mechanics lies a peculiar yet powerful concept: the expectation value. When a quantum system resides in a state described by the wave function $|Ψ⟩$, the expectation value of an observable A is given by the seemingly cryptic expression $⟨Ψ|A|Ψ⟩$. But what lies beneath this mathematical notation? Why does this particular formula accurately predict the average outcome of measuring the observable A on a multitude of identically prepared systems? To unravel this fundamental question, we must delve into the core principles of quantum mechanics, exploring the roles of operators, Hilbert space, observables, and eigenvalues.

Quantum States and Observables: A Primer

To understand the significance of $⟨Ψ|A|Ψ⟩$, let's first lay the groundwork by defining key concepts in quantum mechanics. A quantum system, such as an electron in an atom or a photon in a beam of light, is described by its quantum state, represented by a vector $|Ψ⟩$ in a complex vector space known as Hilbert space. This state vector encapsulates all the information about the system's physical properties. Observables, on the other hand, are the physical quantities that we can measure, such as position, momentum, energy, and angular momentum. In the language of quantum mechanics, each observable is associated with a linear operator A that acts on the state vectors in Hilbert space.

The act of measurement in quantum mechanics is not a passive observation; it fundamentally alters the state of the system. When we measure an observable A, the system is forced into one of the eigenstates of the operator A. Eigenstates are special state vectors that, when acted upon by the operator A, simply get multiplied by a constant factor, called the eigenvalue. Mathematically, this is expressed as:

$A|a⟩ = a|a⟩$

where $|a⟩$ is an eigenstate of A, and a is the corresponding eigenvalue. The eigenvalues represent the possible values that we can obtain when measuring the observable A. For instance, if A represents the energy of an electron in an atom, the eigenvalues would correspond to the allowed energy levels of the electron.

The Superposition Principle: A Quantum Twist

One of the most crucial concepts in quantum mechanics is the superposition principle. It states that if a quantum system can exist in multiple states, it can also exist in a linear combination, or superposition, of those states. This means that the state vector $|Ψ⟩$ can be expressed as a sum of eigenstates of the observable A:

$|Ψ⟩ = c_1|a_1⟩ + c_2|a_2⟩ + ... + c_n|a_n⟩$

where $|a_i⟩$ are the eigenstates of A, and $c_i$ are complex coefficients. The coefficients $c_i$ determine the contribution of each eigenstate to the overall state $|Ψ⟩$. The absolute square of these coefficients, $|c_i|^2$, has a profound physical interpretation: it represents the probability of obtaining the eigenvalue $a_i$ when we measure the observable A on the system in state $|Ψ⟩$. This probabilistic nature is a hallmark of quantum mechanics, distinguishing it from classical physics where measurements are deterministic.

Deriving the Expectation Value Formula: A Journey Through Probabilities

Now, we arrive at the central question: why is the expectation value given by $⟨Ψ|A|Ψ⟩$? The expectation value, by definition, is the average value of the observable A that we would obtain if we performed the measurement on a large number of identically prepared systems. To calculate this average, we need to consider the possible outcomes of the measurement (the eigenvalues $a_i$) and their respective probabilities ($|c_i|^2$).

The expectation value, denoted by $⟨A⟩$, can be calculated as a weighted average of the eigenvalues:

$⟨A⟩ = a_1|c_1|^2 + a_2|c_2|^2 + ... + a_n|c_n|^2 =

∑ᵢ aᵢ|cᵢ|²$

This formula simply states that we sum up the product of each possible value $a_i$ and its probability $|c_i|^2$. Now, let's connect this probabilistic picture to the expression $⟨Ψ|A|Ψ⟩$. Recall that $|Ψ⟩$ can be written as a linear combination of eigenstates:

$|Ψ⟩ = ∑ᵢ cᵢ|aᵢ⟩$

The bra vector $⟨Ψ|$ is the conjugate transpose of $|Ψ⟩$, given by:

$⟨Ψ| = ∑ᵢ cᵢ*⟨aᵢ|$

where $c_i*$ is the complex conjugate of $c_i$.

Now, let's compute $⟨Ψ|A|Ψ⟩$:

$⟨Ψ|A|Ψ⟩ = (∑ᵢ cᵢ*⟨aᵢ|)A(∑ⱼ cⱼ|aⱼ⟩)$

Since A is a linear operator, we can distribute it inside the summation:

$⟨Ψ|A|Ψ⟩ = ∑ᵢ ∑ⱼ cᵢ*cⱼ⟨aᵢ|A|aⱼ⟩$

We know that $A|a_j⟩ = a_j|a_j⟩$, so:

$⟨Ψ|A|Ψ⟩ = ∑ᵢ ∑ⱼ cᵢ*cⱼ⟨aᵢ|aⱼ⟩aⱼ$

The eigenstates of an observable are orthonormal, meaning that their inner product is zero unless they are the same state:

$⟨aᵢ|aⱼ⟩ = δᵢⱼ$

where $δᵢⱼ$ is the Kronecker delta, which is 1 if i = j and 0 otherwise. This simplifies the expression to:

$⟨Ψ|A|Ψ⟩ = ∑ᵢ ∑ⱼ cᵢ*cⱼδᵢⱼaⱼ = ∑ᵢ cᵢ*cᵢaᵢ = ∑ᵢ aᵢ|cᵢ|²$

This is precisely the same formula we derived for the expectation value based on probabilistic arguments! Therefore, we have shown that:

$⟨Ψ|A|Ψ⟩ = ⟨A⟩$

The Significance of $⟨Ψ|A|Ψ⟩$: A Bridge Between Theory and Experiment

The formula $⟨Ψ|A|Ψ⟩$ is a cornerstone of quantum mechanics, providing a direct link between theoretical predictions and experimental observations. It tells us that the expectation value of an observable A in a state $|Ψ⟩$ can be calculated by