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Observables

Abstract

The details of how measurements on quantum states are modelled.

Keywords:MeasurementsHermitian OperatorsEigenvaluesEigenvectors.

Definition

An observables is a physical property that can be measured.

In quantum mechanics, observables are modelled by hermitian operators.

In the following all the concepts needed to understand this definition and why it is used will be built up.

Hermitian Operators

The adjoint of a linear operator AA, denoted by A†A^{\dagger}, is defined by

⟨ψ∣A†∣ϕ⟩=⟨ϕ∣A∣ψ⟩∗,\begin{align*} \bra{\psi} A^{\dagger} \ket{\phi} = \bra{\phi} A \ket{\psi}^{*}, \end{align*}
(1)

where (⋅)∗(\cdot)^* is a the complex conjugate and (⋅)†(\cdot)^{\dagger} is the adjoint operator and ∣ψ⟩,∣ψ⟩\ket{\psi}, \ket{\psi} are general states. See here for more properties of the adjoint operator.

It can be seen from this definition that the matrix of A†A^{\dagger} is the matrix of AA transposed with elements complex conjugated.

Details of This Definition

Let (⋅,⋅):Cn⊗Cn→C1(\cdot, \cdot): \mathbb{C}^{n} \otimes \mathbb{C}^{n} \rightarrow \mathbb{C}^{1} be an inner product, and A:H→HA: \mathcal{H} \rightarrow \mathcal{H} a linear operator. The adjoint of the linear operator AA, denoted by A†A^{\dagger}, is the operator that satisfies the following relation

(A†∣ψ⟩,∣ϕ⟩)=(∣ψ⟩,A∣ϕ⟩),(A^{\dagger} \ket{\psi}, \ket{\phi}) = (\ket{\psi}, A \ket{\phi}),
(2)

where ∣ψ⟩,∣ϕ⟩ ∈ H\ket{\psi}, \ket{\phi}~\in~\mathcal{H}. Explicitly inputting the form of the inner product we are using gives

[⟨ψ∣A†]∣ϕ⟩=⟨ψ∣[A∣ϕ⟩],\begin{align*} \big[ \bra{\psi} A^{\dagger} \big] \ket{\phi} = \bra{\psi} \big[ A \ket{\phi} \big], \end{align*}
(3)

where the brackets are used to show which ket the operator is acting upon. Letting

∣ϕ′⟩=A∣ϕ⟩,\ket{\phi'} = A \ket{\phi},
(4)

gives

[⟨ψ∣A†]∣ϕ⟩=⟨ψ∣ϕ′⟩,=⟨ϕ′∣ψ⟩∗,=[⟨ϕ∣A]∣ψ⟩∗.\begin{align*} \big[ \bra{\psi} A^{\dagger} \big] \ket{\phi} &= \braket{\psi|\phi'}, \\ &= \braket{\phi'|\psi}^{*}, \\ & = \big[ \bra{\phi} A] \ket{\psi}^{*}. \end{align*}
(5)

where the properties of the inner product have been used. Dropping the brackets gives the definition of the adjoint given in the main text.

An operator AA is said to be Hermitian, or self-adjoint, if

A†=A.A^\dagger = A.
(6)

Properties of Hermitian Operators

  1. Hermitian operators have real eigenvalues.
  2. The eigenvectors of hermitian operators with different eigenvalues are orthogonal.
  3. The eigenvectors of hermitian operators form a complete set of basis states.
Proofs of (most) of these properties

Let AA be a hermitian operators acting on a Hilbert space.

  1. Hermitian operators have real eigenvalues: The eigenvalue equation in terms of bra-ket notion is
A∣a⟩=a∣a⟩,A \ket{a} = a \ket{a},
(7)

where aa is the eigenvalue of the eigenvector ∣a⟩\ket{a}.

Inputting an eigenvector of AA into the definition of the adjoint with a hermitian operator gives

⟨a∣A∣a⟩=⟨a∣A∣a⟩∗,\begin{align*} \bra{a} A \ket{a} = \bra{a} A \ket{a}^{*}, \end{align*}
(8)

as A†=AA^\dagger = A.

The left hand side of this equation gives

⟨a∣A†∣a⟩=⟨a∣a∣a⟩,=a⟨a∣a⟩,=a.\begin{align*} \bra{a} A^{\dagger} \ket{a} &= \bra{a} a \ket{a}, \\ &= a \braket{a|a}, \\ &= a. \end{align*}
(9)

The right hand side of this equation gives

⟨a∣A∣a⟩∗=⟨a∣a∣a⟩∗,=a∗⟨a∣a⟩∗,=a∗.\begin{align*} \bra{a} A \ket{a}^* &= \bra{a} a \ket{a}^*, \\ &= a* \braket{a|a}^*, \\ &= a^*. \end{align*}
(10)

Hence, a=a∗a=a^* meaning aa must be a real number.

  1. The eigenvectors of hermitian operators with different eigenvalues are orthogonal: Consider two eigenvectors of AA given by
A∣a⟩=a∣a⟩  and  A∣a′⟩=a′∣a′⟩,A \ket{a} = a \ket{a}~~ \textrm{and} ~ ~ A \ket{a'} = a' \ket{a'},
(11)

such that a≠a′a \neq a'. Inputting them into the definition of the adjoint gives

⟨a′∣A∣a⟩=⟨a∣A∣a′⟩∗,\begin{align*} \bra{a'} A \ket{a} = \bra{a} A \ket{a'}^{*}, \end{align*}
(12)

which can be rewritten as

(a−a′)⟨a′∣a⟩=0.\begin{align*} (a-a')\braket{a'|a} = 0. \end{align*}
(13)

Given a≠a′a \neq a' it must be the case that ⟨a′∣a⟩=0\braket{a'|a} = 0.

  1. The eigenvectors of hermitian operators form a complete set of basis states: This result is given by the spectral theorem which says that an operator AA acting on Hd\mathcal{H}^d is normal, meaning A†A=AA†A^\dagger A = A A^\dagger (which is satisfied by all hermitian operators), if and only if it is diagonalisable in some othronormal basis.

Hence, AA can be written as

A=∑i=0d−1λi∣λi⟩⟨λi∣,A = \sum^{d-1}_{i=0} \lambda_i \ket{\lambda_i}\bra{\lambda_i},
(14)

where

A∣λi⟩=λi∣λi⟩ ∀ iA \ket{\lambda_i} = \lambda_i \ket{\lambda_i} ~ \forall~i
(15)

and

⟨i∣j⟩=δij ∀ i,j.\braket{i|j} = \delta_{ij}~\forall~i,j.
(16)

This means AA can always be written as a diagonal operator with respect to it’s orthonormal eigenbasis.

It could be the case that λi=λj\lambda_i = \lambda_j for some values of i,ji,j. In this case the eigenvalues are said to be degenerate. In this case, AA can be written as

A=∑iλiPi,A = \sum_{i} \lambda_i P_{i},
(17)

where PiP_{i} is the projection onto the eigenspace with eigenvalue λi\lambda_i (meaning the Hilbert space spanned by the set of eigenvectors with eigenvalue λi\lambda_i). Hence, if

A∣κj⟩=λi∣κj⟩A \ket{\kappa_j} = \lambda_i \ket{\kappa_j}
(18)

for all vectors in the set {∣κj⟩}jk≤d\{ \ket{\kappa_j} \}_j^{k\leq d} then

Pi=∑jk∣κj⟩⟨κj∣.P_{i} = \sum_j^{k} \ket{\kappa_j}\bra{\kappa_j}.
(19)

Measuring Observables

The possible outcomes of measuring an observables AA are the eigenvalues of AA.

After measuring the observables AA and getting the eigenvalue λ \lambda , for example, the state the measurement was performed on will be in the eigenspace spanned by the eigenvectors with eigenvalue aa.

Quantum mechanics does not tell us which outcome will occur as a result of measuring an operator, it only tells us the probabilities of getting each possible outcome. After measuring AA on a state, the output is a random variable where the outcome is λ \lambda with some probability P(λ)P(\lambda).

Let AA be an observable such that

A=∑iλiPi,A = \sum_{i} \lambda_i P_{i},
(20)

where PiP_i is the projection onto the eigenspace of AA with eigenvalue λi\lambda_i, and ∣ψ⟩∈H\ket{\psi}\in\mathcal{H}.

Probabilities of Measurement Outcomes:

The probability of measuring the observable AA on ∣ψ⟩\ket{\psi} and getting the outcome λi\lambda_i is

P(λi∣A)=⟨ψ∣Pi∣ψ⟩.P(\lambda_i \vert A) = \bra{\psi} P_i \ket{\psi}.
(21)

If the eigenspace of λi\lambda_i is not dengerate, then PiP_i is just the projection onto the eigenstate of AA with eigenvalue λi\lambda_i, meaning Pi=∣λi⟩⟨λi∣P_i = \ket{\lambda_i}\bra{\lambda_i}, and hence,

P(λi∣A)=∣⟨λi∣ψ⟩∣2.P(\lambda_i | A) = \vert \braket{\lambda_i|\psi} \vert ^ 2.
(22)

Let {λi}i=0d−1\{ \lambda_i \}_{i=0}^{d-1} be the eigenvectors of a hermitain operator AA. Given this set of vectors forms an orthonormal basis a state ∣ψ⟩ \ket{\psi} can be decomposed in this basis,

∣ψ⟩=∑ipi∣λi⟩,\begin{align*} \ket{\psi} = \sum_i p_i \ket{\lambda_i}, \end{align*}
(23)

where pi=⟨λi∣ψ⟩p_i = \braket{\lambda_i|\psi}. It can therefore be seen that ∣pi∣2\vert p_i \vert ^2 gives the probability of measuring AA on ∣ψ⟩\ket{\psi} and getting the outcome λi\lambda_i.

Therefore, when considering measuring the observable AA on a state ψ\psi, one should first decompose the state ∣ψ⟩\ket{\psi} in the eigenbasis of AA. The coefficients of the decomposition can then be used to find the probabilities of getting the different measurement outcomes. Note, when measuring the observable AA one should not apply the observable to the state.

Post Measurement State:

After measuring an observable AA on the state ∣ψ⟩\ket{\psi} and getting the outcome λi\lambda_i the post measurement state, ∣ψpost⟩\ket{\psi_{\rm post}}, is the state ∣ψ⟩\ket{\psi} projected into the eigenspace of the AA with eigenvalue λi\lambda_i,

∣ψpost⟩=Pi∣ψ⟩∣∣Pi∣ψ⟩∣∣2,=Pi∣ψ⟩P(λi∣A)\begin{align*} \ket{\psi_{\rm post}} &= \frac{P_{i} \ket{\psi}}{ \vert \vert P_i \ket{\psi} \vert \vert_2}, \\ &= \frac{P_{i} \ket{\psi}}{ \sqrt{P(\lambda_i | A)} } \end{align*}
(24)

where ∣∣⋅∣∣2 \vert \vert \cdot \vert \vert_2 is the 2-norm and ensures the post measurement state is normalised - and hence a valid state. The second line can be shown as

∣∣Pi∣ψ⟩∣∣2=∣Pi∣ψ⟩∣,=(⟨ψ∣Pi†)(Pi∣ψ⟩),=(⟨ψ∣Pi)(Pi∣ψ⟩),=⟨ψ∣Pi∣ψ⟩,=P(λi∣A),\begin{align*} \vert \vert P_i \ket{\psi} \vert \vert_2 &= \sqrt{ \vert P_i \ket{\psi} \vert }, \\ &= \sqrt{ \big( \bra{\psi} P_i^{\dagger} \big) \big(P_i \ket{\psi} \big) }, \\ &= \sqrt{ \big( \bra{\psi} P_i \big) \big(P_i \ket{\psi} \big) }, \\ &= \sqrt{ \bra{\psi} P_i \ket{\psi} }, \\ &= \sqrt{ P(\lambda_i | A) }, \end{align*}
(25)

where we have used the fact that projection operators onto eigenspaces of hermitian operators are hermitian, Pi†=PiP_i^{\dagger}=P_i and Pi2=PiPi=PiP_i^{2} = P_iP_i = P_i.

Expectation Values:

The expectation value of an observable is the average outcome one would expect if measuring AA on many copies of a state,

⟨A⟩=∑iλiP(λi∣A),\begin{align*} \braket{A} = \sum_i \lambda_i P(\lambda_i \vert A), \end{align*}
(26)

which is the probability of getting the outcome λi\lambda_i multiplied by the outcome itself, summed over all possible outcomes from measuring AA.

The expectation value of AA on ∣ψ⟩\ket{\psi} has a succinct form as

⟨A⟩=⟨ψ∣A∣ψ⟩,=⟨ψ∣(∑iλiPi)∣ψ⟩,=∑iλi⟨ψ∣Pi∣ψ⟩,=∑iλiP(λi∣A).\begin{align*} \braket{A} &= \bra{\psi} A \ket{\psi}, \\ &= \bra{\psi} \biggl( \sum_{i} \lambda_i P_{i} \biggl) \ket{\psi}, \\ &= \sum_i \lambda_i \bra{\psi} P_i \ket{\psi}, \\ &= \sum_i \lambda_i P(\lambda_i \vert A). \end{align*}
(27)

Higher moments of expectation values can also be found as

⟨A2⟩=⟨ψ∣A2∣ψ⟩,=∑iλi2P(λi∣A),\begin{align*} \braket{A^2} &= \bra{\psi} A^2 \ket{\psi}, \\ &= \sum_i \lambda_i^2 P(\lambda_i \vert A), \end{align*}
(28)

which generalises to the nnth moment as expected.

From this, the standard deviation can be found as

σA=⟨A2⟩−⟨A⟩2,=⟨(A−⟨A⟩)2⟩,\begin{align*} \sigma_{A} &= \sqrt{\braket{A^2} - \braket{A}^2}, \\ &= \sqrt{ \braket{ (A - \braket{A} )^2 } }, \end{align*}
(29)

which can be interpreted as the standard deviation of the measurement outcomes if the observable AA is measured on many copies of ∣ψ⟩\ket{\psi}.

The Uncertainty Principle

The Heisenberg uncertainty principle sets a fundamental limit on the accuracy to which two non-commuting observables can be known. It comes in a few different forms, with the Robertson form being most widely used.

Let AA and BB be two observables. The Robertson form of the uncertainty principle is:

σAσB≥12∣⟨[A,B]⟩∣,\sigma_A \sigma_B \geq \frac{1}{2} \vert \braket{ \big[A,B\big]} \vert,
(30)

where σA,σB\sigma_A, \sigma_B are the standard deviations of the measurement outcomes of the observables AA and BB respectively, and [A,B]=AB−BA[A,B] = AB-BA is the commutator of the observables.

One should interpret this as preparing some number of copies of a state ∣ψ⟩\ket{\psi}, and measuring AA on half the copies, and BB on the other half of the copies. The product of the standard deviation of the AA measurement outcomes and the standard deviation of the BB measurement outcomes, is what the uncertainty principle bounds.

Note, the expectation value included in the above definition of the uncertainty principle is a state dependent quantity. Hence, the limitation on the accuracy imposed by the uncertainty principle is dependent on the state, ∣ψ⟩\ket{\psi}, upon which the measurements are being made.

If two observables commute, meaning [A,B]=0[A,B]=0, then there is no limit to the accuracy with which the two observables can be simultaneously known.

See here for some details on commutators

Why Hermitian Operators as Observables

In this section we will aim to motivate why hermitian operators should be used to represent things that can be measured in quantum mechanics. Below are some reasons commonly found in the literature.

  1. They give real measurement outcomes: as proved above, the eigenvalues of Hermitian matrices are real. Given the eigenvalues of an observables are the possible measurement outcomes one can get, this means that all possible measurement outcomes, for all possible things that can be measured, are real. In The Principles of Quantum Mechanics, Dirac argues that all possible measurement outcomes must be real due to the potential for the measurement of an observables to alter the state (e.g the post measurement state becomes an eigenstate of the observable measured). If one were to try and measure a complex number associated to a quantum state, they would need two real numbers to specify it. The observer could try and measure the real and imaginary part separately. However, in general, performing a measurement to extract the real part will alter the state, changing what one would get if they then performed a measurement to extract the imaginary part. For this reason, Dirac claims that all measurement outcomes must be real.

  2. Retains probability: this refers to the eigenvectors of Hermitian operators forming a complete basis, as stated by the spectral theorem. This ensures that the sum of the probabilities of getting the different possible measurement outcomes always sums to one. To see this, let A=∑iλiPiA = \sum_i \lambda_i P_i. Given AA is Hermitian, ∑Pi=I\sum P_i = \mathbb{I}. Hence, the sum over the probabilities of the different possible measurement outcomes of measuring AA on ∣ψ⟩\ket{\psi} is

∑i⟨ψ∣Pi∣ψ⟩=⟨ψ∣(∑iPi)∣ψ⟩,=⟨ψ∣ψ⟩,=1\begin{align*} \sum_i \bra{\psi} P_i \ket{\psi} &= \bra{\psi} \bigg( \sum_i P_i \bigg) \ket{\psi}, \\ &= \braket{\psi|\psi}, \\ &= 1 \end{align*}
(31)

Therefore, all possibilities are properly accounted for, when measuring AA on a normalised state ∣ψ⟩\ket{\psi}, one will get a measurement outcome associated to AA with certainty. If the sum was less then 1, then there would be some probability “unaccounted” for.

Another question one might have is

Why are post-measurement states are eigenvectors of an observables: it is easy to see from the framework presented above that if one has a state that is an eigenvector of an observable, and that observable is then measured on the state, that one will get the same outcome each time. Namely, they will get the eigenvalue associated to that eigenvector. But, is the converse true? Is a state that always gives the same outcome when an observable is measured on it necessarily an eigenvector of the observable being measured? The answer, proved below, is yes.

Proof

Consider one has multiple copies of a state ∣ψ⟩\ket{\psi} and on each copy they measure the observable AA and get the real outcome a∈R1a \in \mathbb{R}^1.

The standard deviation of the measurement is zero, σ=0\sigma=0, and the average is aa, ⟨A⟩=a\braket{A}=a.

Therefore,

σ2=⟨(A−⟨A⟩)2⟩,=⟨ψ∣(A−a)2∣ψ⟩,=⟨ψ∣[(A−a)(A−a)∣ψ⟩],=[⟨ψ∣(A−a)†][(A−a)∣ψ⟩],=[⟨ψ∣(A−a)][(A−a)∣ψ⟩],\begin{align*} \sigma^2 &= \braket{ (A - \braket{A})^2 }, \\ &= \bra{\psi} (A - a)^2 \ket{\psi}, \\ &= \bra{\psi} \big[ (A-a)(A-a)\ket{\psi} \big], \\ &= \big[ \bra{\psi} (A-a)^{\dagger} \big] \big[ (A-a)\ket{\psi} \big], \\ &= \big[ \bra{\psi} (A-a) \big] \big[ (A-a)\ket{\psi} \big], \end{align*}
(32)

The adjoint of the operator is taken so that (A-a) now acts on the bra, but A−aA-a is Hermitian.

The above is zero if and only if

(A−a)∣ψ⟩=0,(A-a)\ket{\psi} = 0,
(33)

as the only vector who’s inner product with itself is zero is the zero vector.

Hence,

A∣ψ⟩=a∣ψ⟩,A \ket{\psi} = a \ket{\psi},
(34)

which is the eigenvalue equations, proving that a state with a definitive outcome when measured by an observables is an eigenvector of that observable.

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