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Chapter 1a: Principles of Quantum Mechanics

1.1. Hilbert spaces:

Hilbert Space:

  • Hilbert space is a vector space \(\mathcal{H}\) over \(\mathbb{C}\) that is equipped with a complete inner product.
  • This definition has 3 keywords:
    • Vector space - is well known,
    • Complete - is just a hedge against infinite Hilbert Spaces (unimportant)
    • Inner Product is a map \((\quad,\quad): \mathcal{H} \times \mathcal{H} \rightarrow \mathbb{C}\) that obeys
\[ \begin{aligned} \text { conjugate symmetry } & (\phi, \psi)=\overline{(\psi, \phi)} \\ \text { linearity } & (\phi, a \psi)=a(\phi, \psi) \\ \text { additivity } & (\phi, \psi+\chi)=(\phi, \psi)+(\phi, \chi) \\ \text { positive-definiteness } & (\psi, \psi) \geq 0 \forall \psi \in \mathcal{H} \end{aligned} \]
  • Metric of \(\mathcal{H}\) is defined by the norm \(\|\psi\| \equiv \sqrt{(\psi, \psi)}\)
  • Quantum states can be represented as a vectors \(\psi \in \mathcal{H}\)

Dual Spaces:

  • Dual space \(\mathcal{H^*}\) of a \(\mathcal{H}\) is the space of linear maps \(\mathcal{H} \rightarrow \mathbb{C}\). That is, an element \(\phi \in \mathcal{H^*}\) defines a map \(\varphi: \psi \mapsto \varphi(\psi) \in \mathbb{C}\) for every \(\psi \in \mathcal{H}\), such that
\[ \varphi: a \psi_1+b \psi_2 \mapsto a \varphi\left(\psi_1\right)+b \varphi\left(\psi_2\right) \]
  • One of the dual space \(\mathcal{H^*}\) is for instance the inner product \((\phi, \quad) \in \mathcal{H}^*\) for \(\phi \in \mathcal{H}\), where
\[ (\phi, \quad): \psi \mapsto(\phi, \psi) \]

1.2. Dirac Notation:

In quantum mechanics quite often we often switch basis. This is because intrinsically any measurement causes a collapse onto the measurement basis. Because of this we want to have a notation that allows us to work with multiple basis at the same time, and not get confused. Dirac notation (empirically) provides this clarity. It is difficult to formally define the notation, and quite often when one does it, they get confused (unless they are deep down in theory). Therefore I would propose to learn it through learning the basic few properties and then trying things out.

Dirac denotes element of \(\mathcal{H}\) as \(\left|\psi\right>\) 'ket', and an element of the dual space is written as \(\mathcal{H^*}\) as \(\left<\psi\right|\) 'bra'. The inner product between two states \(\left|\psi\right>, \left|\phi\right> \in \mathcal{H}\) is written as \(\left<\psi|\phi\right>\).

In notes the bra-ket notation is introduced using homomorphisms (linear maps). I find it unecessary.

  • The advantage of using bra-ket notation is:
    • We can talk about multiple things at the same time - Dirac notation is effectively just a label that points to an abstract object in the Hilbert space. We don't need to specify whether the variable is contineous, or if it is a vector or a function.
    • Allows us to label states by their eigenvalues
    • Somehow it is more natural and causes less confusion

1.3. Operators:

  • A linear operator A is a map \(A : \mathcal{H} \rightarrow \mathcal{H}\) that is compatible with the vector space structure \(A(c_1\left|\phi_1\right> + c_2\left|\phi_2\right>) = c_1A\left|\phi_1\right> + c_2A\left|\phi_2\right>\)
  • All operators in Quantum Mechanics are linear, hence we will call them just 'operators'
  • Operators form algebra
    • Sum: \((\alpha A+\beta B):\left|\phi\right> \mapsto \alpha A\left|\phi\right>+\beta B\left|\phi\right>\)
    • Product: \(A B: \phi \mapsto A \circ B\left|\phi\right>=A(B\left|\phi\right>)\)
    • Commutator: \([A, B]=A B-B A\)
  • A state \(\psi \in \mathcal{H}\) is said to be an eigenstate of an operator A if \(A\left|\psi\right> = a_\psi\left|\psi\right>\) with an associated eigenvalue '\(a_\psi\)'.
  • Adjoint \(A^\dagger\) of an operator \(A\) is defined as \(\left<\phi\right|A^{\dagger}\left| \psi\right>=\overline{\left<\psi\right|A\left| \phi\right>} \quad\)
  • An operator \(Q\) is called Hermitian if \(Q^\dagger=Q\)
  • An operator \(U\) is called Unitary if \(U^\dagger U= U U^\dagger = \mathbb{I}\)
  • An operator \(\Pi\) is called Projector if \(\Pi\Pi= \Pi\)

1.4. Composite systems:

Tensor Product

  • Tensor product \(\mathcal{H}_1 \otimes \mathcal{H}_2\) is a vector space over \(\mathbb{C}\) spanned by all pairs of elements \(\left|e_a\right> \otimes\left|f_\alpha\right>\), where \(\left|e_a\right> \in \mathcal{H_1}\), \(\left|f_\alpha\right> \in \mathcal{H_2}\)
  • It is not true that a general element of \(\mathcal{H}_1 \otimes \mathcal{H}_2\) necessarily takes the form \(\left|\psi_1\right>\otimes\left|\psi_2\right>\)
  • Rahter, a general element may be written as \(\left|\Psi\right>=\sum_{a, \alpha} r_{a \alpha}\left|e_a\right> \otimes\left|f_\alpha\right>\)
  • Elements of the form \(\left|\psi_1\right>\otimes\left|\psi_2\right>\) are called simple, and the elements of the form \(\left|\Psi\right>=\sum_{a, \alpha} r_{a \alpha}\left|e_a\right> \otimes\left|f_\alpha\right>\) are refered as entangled
  • \(\left<\alpha\otimes\beta|\alpha'\otimes\beta'\right> := \left<\alpha|\alpha'\right>\left<\beta|\beta'\right>\)
  • \(\left( S_\alpha \otimes T_\beta \right)\left(\alpha \otimes \beta\right) = \left(S_\alpha\alpha\right)\otimes\left(T_\beta\beta\right)\) - apologies for being slightly sloppy - I think it is understandable what I mean though

Tensor Product in action (states)

  • Let's as an example consider that our states \(\left|\alpha\right>_A \text{ and } \left|\beta\right>_B\) live both in \(\mathbb{C}^2_A\) and \(\mathbb{C}^2_B\) respectively. Then we can pick orthonormal basis of \(\mathbb{C}^2_A\) to be \(\left\{\left|u_1\right>_A, \left|u_2\right>_A \right\}\), and of \(\mathbb{C}^2_B\) to be \(\left\{\left|v_1\right>_B, \left|v_2\right>_B \right\}\)
  • Then one can write \(\left|\alpha\right>_A = a_1 \left|u_1\right>_A + a_2 \left|u_2\right>_A = a_1 \begin{pmatrix} 1\\ 0 \end{pmatrix}_A + a_2 \begin{pmatrix} 0\\ 1 \end{pmatrix}_A\),
  • and \(\left|\beta\right>_B = b_1 \left|v_1\right>_B + b_2 \left|v_2\right>_B = b_1 \begin{pmatrix} 1\\ 0 \end{pmatrix}_B + b_2 \begin{pmatrix} 0\\ 1 \end{pmatrix}_B\)
  • This means that one can write
\[ \left|\alpha\right>_A \otimes \left|\beta\right>_B = \begin{pmatrix} a_1\\ a_2 \end{pmatrix}_A \otimes \begin{pmatrix} b_1\\ b_2 \end{pmatrix}_B = \begin{pmatrix} a_1b_1\\ a_1b_2\\ a_2b_1\\ a_2b_2 \end{pmatrix} \]
  • or sticking to the Dirac notation:
\[ \left|\alpha\right>_A \otimes \left|\beta\right>_B = \sum_{i,j} a_i b_j \left|u_i\right>_A \otimes \left|v_j\right>_B \]

Tensor Product in action (operators)

  • For operators \(A\) and \(B\) that live in \(\mathbb{C}^2_A\) and \(\mathbb{C}^2_B\) respectively, one can write \(A = \sum_{i,j} a_{ij} \left|u_i\right>_A \left<u_j\right|\) and \(B = \sum_{i,j} b_{ij} \left|v_i\right>_B \left<v_j\right|\)
  • This means:
\[ A \otimes B = \sum_{i,j,k,\ell} a_{ij} b_{k\ell} \left|u_i\right>_A \otimes \left|v_k\right>_B \left<u_j\right|\otimes\left<v_\ell\right| \]
  • or in a matrix form:
\[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B\\ a_{21}B & a_{22}B \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} & a_{11}b_{12} & a_{12}b_{11} & a_{12}b_{12}\\ a_{11}b_{21} & a_{11}b_{22} & a_{12}b_{21} & a_{12}b_{22}\\ a_{21}b_{11} & a_{21}b_{12} & a_{22}b_{11} & a_{22}b_{12}\\ a_{21}b_{21} & a_{21}b_{22} & a_{22}b_{21} & a_{22}b_{22}\\ \end{pmatrix} \]

1.5. Postulates of Quantum Mechanics:

  • (1) A quantum system A is associated with complex Hilber space \(\mathcal{H}\). A physical state of an isolated system is represented by a normalised vector \(\left|\psi\right> \in \mathcal{H}\), which is unique up to a phase factor
  • (2) The evolution of an isolated quantum system is reversible. In this formalism this corresponds to unitary evolution of the form \(\left|\psi\right> \mapsto U\left|\psi\right>\) for \(U \in \mathcal{U}(\mathcal{H})\), i.e. \(U^{\dagger} U=U U^{\dagger}=\mathbb{I}\). The unitary is unique up to a phase factor
  • (3) Composite system - For two quantum system A, and B with associated Hilber spaces \(\mathcal{H_A}\) and \(\mathcal{H_B}\) the Hilbert space \(\mathcal{H_{AB}}\) associated with the composite system AB is isomorphic to the tensor product \(\mathcal{H_A}\otimes\mathcal{H_B}\). For unitary operation on the subsystem we use: \(U_A \otimes \mathbb{I}_B\left|i j\right>_{A B} \equiv U_A\left|i j\right>_{A B}\)
  • (4) Measurement - A projective measurement on a quantum system with outcomes labelled \({x}_x\) is associated with a set of projectors \({\Pi_x}x\) satisfying \(\sum_x \Pi_x = \mathbb{I}\).
    • Probability of getting outcome x when measuring state \(\left|\psi\right>\) is given by the Born rule: \(Pr[x \mid \psi]=\left\langle\psi\left|\Pi_x\right| \psi\right\rangle\)
    • Post-measurement state is given the outcome x is \(\left|\psi_x^{\prime}\right>=\frac{1}{\sqrt{\operatorname{Pr}[x \mid \psi]}} \Pi_x\left|\psi\right>=\frac{\Pi_x\left|\psi\right>}{\sqrt{\left\langle\psi\left|\Pi_x\right| \psi\right\rangle}}\)