Math Infinitum

Mapping the lesson.

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Intuition

Once a basis is chosen, a state is nothing but its list of coefficients, one amplitude per basis state. The coefficient of a basis state is the bracket of that state with the one being described, the modulus squared of each coefficient is a probability, and the probabilities add to one because the state is normalised. Inner products and norms can then be computed from the lists alone.

A recipe lists how much of each ingredient goes in. Choose a different set of ingredients and the same dish needs a different list, but it is the same dish. The coefficients are the recipe of a state in one basis.

The moduli squared $|c_{n}|^{2}$ of one normalised state in a basis of five states. They are the probabilities of the five outcomes, and they add to one: $0.1+0.4+0.3+0.15+0.05=1$ .

Coefficients in a basis

In an orthonormal basis $\{\ket{n}\}$ a state is $\ket{\psi}=\sum_{n}c_{n}\ket{n}$ with . Its norm and its inner products with other states are computed from the coefficients alone, and for a normalised state the numbers are the probabilities of finding it in each basis state.

c_{n}=\braket{n|\psi},\qquad \braket{\psi|\psi}=\sum_{n}|c_{n}|^{2}

Computing with coefficients

The inner product of $\ket{\phi}=\sum b_{n}\ket{n}$ with $\ket{\psi}=\sum c_{n}\ket{n}$ is , in any orthonormal basis.

Parseval’s identity

\braket{\psi|\psi}=\sum_{n}|\braket{n|\psi}|^{2}

Insert the completeness relation between the bra and the ket of $\braket{\psi|\psi}$ . Each term of the resulting sum is a bracket times its own conjugate, a modulus squared. The length of a vector is fully accounted for by its components along any orthonormal basis.

Proof steps

\braket{\psi|\psi}=\bra{\psi}\,\hat{1}\,\ket{\psi}

Nothing changes when the identity is put between the bra and the ket.

=\sum_{n}\braket{\psi|n}\braket{n|\psi}

Write the identity as $\sum_{n}\ket{n}\bra{n}$ , the completeness relation.

\braket{\psi|n}=\braket{n|\psi}^{*}=c_{n}^{*}

Swapping the sides of a bracket conjugates it.

\braket{\psi|\psi}=\sum_{n}c_{n}^{*}c_{n}=\sum_{n}|c_{n}|^{2}

Each term is a coefficient times its conjugate, its modulus squared.

Applications

Practice

Coefficients Are Amplitudes

In a basis, a state is its list of coefficients. For a normalised state, the modulus squared of each coefficient is the probability of the corresponding basis state.

P(n)=|c_{n}|^{2}=|\braket{n|\psi}|^{2}

Try it

$\ket{\psi}=\tfrac{1}{\sqrt5}\bigl(\ket{0}+2i\ket{1}\bigr)$ . What is the probability of finding it in $\ket{1}$ ?