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Commit 8716f928 authored by Maciej Topyla's avatar Maciej Topyla
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fix admonitions

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1 merge request!171st major update to lecture note 4
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...@@ -51,27 +51,27 @@ This vector space is known as the *state space* of the system. ...@@ -51,27 +51,27 @@ This vector space is known as the *state space* of the system.
The set of all possible state vectors describing a given physical system forms a complex vector space $\mathcal{H}$, which is known as the *Hilbert space* of the system. You can think of the Hilbert space as the space populated by all possible states that a quantum system can be found on. Hilbert spaces inherit a number of the important properties of general vector spaces: The set of all possible state vectors describing a given physical system forms a complex vector space $\mathcal{H}$, which is known as the *Hilbert space* of the system. You can think of the Hilbert space as the space populated by all possible states that a quantum system can be found on. Hilbert spaces inherit a number of the important properties of general vector spaces:
!!! info "" !!! info "Superposition"
A linear combination (or superposition) of two or more state vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$, is also a state of the quantum system. Therefore, a linear combination $|{\Psi}\rangle$ of the form $$|{\Psi}\rangle=c_1|{\psi_1}\rangle+c_2|{\psi_1}\rangle+c_3|{\psi_3}\rangle+...+c_n|{\psi_n}\rangle = \sum_{i=1}^n c_i|{\psi_i}\rangle$$ A linear combination (or superposition) of two or more state vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$, is also a state of the quantum system. Therefore, a linear combination $|{\Psi}\rangle$ of the form $$|{\Psi}\rangle=c_1|{\psi_1}\rangle+c_2|{\psi_1}\rangle+c_3|{\psi_3}\rangle+...+c_n|{\psi_n}\rangle = \sum_{i=1}^n c_i|{\psi_i}\rangle$$
where $c_1, c_2, c_3, ...$ are general complex numbers will also be a physically allowed state vector of the quantum system. where $c_1, c_2, c_3, ...$ are general complex numbers will also be a physically allowed state vector of the quantum system.
!!! info "" !!! info "Normalisation"
If a physical state of the system is given by a vector $|{\Psi}\rangle$, then the same physical state can also be represented by the vector $c|{\Psi}\rangle$ where $c$ is a non-zero complex number. The reason for this is that the overall normalisation of the state vector *does not change the physics* of the system (or in other words, does not modify the *information content* of the state vector). As we will discuss below, in quantum mechanics it is advantageous to work with *normalised vectors*, that is, whose *length* is one. If a physical state of the system is given by a vector $|{\Psi}\rangle$, then the same physical state can also be represented by the vector $c|{\Psi}\rangle$ where $c$ is a non-zero complex number. The reason for this is that the overall normalisation of the state vector *does not change the physics* of the system (or in other words, does not modify the *information content* of the state vector). As we will discuss below, in quantum mechanics it is advantageous to work with *normalised vectors*, that is, whose *length* is one.
We will define in a while what do we mean by length. We will define in a while what do we mean by length.
!!! info "" !!! info "Completeness"
A set of vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$ is said to be *complete* if every state A set of vectors $|{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle$ is said to be *complete* if every state
of the quantum system can be represented as a linear combination of them. of the quantum system can be represented as a linear combination of them.
In such a case, it becomes possible to express *any* state vector $|{\Psi}\rangle$ of the system's Hilbert space as a superposition of these $n$ vectors, In such a case, it becomes possible to express *any* state vector $|{\Psi}\rangle$ of the system's Hilbert space as a superposition of these $n$ vectors,
$$ |{\Psi}\rangle=\sum_{i=1}^n c_i|{\psi_i}\rangle$$ $$ |{\Psi}\rangle=\sum_{i=1}^n c_i|{\psi_i}\rangle$$
for some specific choice of coefficients $c_i$. The set of vector \{$|{\psi_i}\rangle$\} are then said to *span* the Hilbert space of the quantum system. for some specific choice of coefficients $c_i$. The set of vector \{$|{\psi_i}\rangle$\} are then said to *span* the Hilbert space of the quantum system.
!!! info "" !!! info "Basis"
A set of vectors \{$|{\psi_i}\rangle$\} is said to form a basis for the state space if the set of vectors is *complete* and if in addition they are *linearly independent*. The latter condition means essentially that one cannot express a given basis vector as a linear combination of the rest of basis vectors. A set of vectors \{$|{\psi_i}\rangle$\} is said to form a basis for the state space if the set of vectors is *complete* and if in addition they are *linearly independent*. The latter condition means essentially that one cannot express a given basis vector as a linear combination of the rest of basis vectors.
Linear independence can also be expressed as the requirement that if one has that Linear independence can also be expressed as the requirement that if one has that
$$\sum_{i=1}^n c_i |{\psi_i}\rangle=0\;\text{then}\; c_i=0\;\text{for all}\; i$$ $$\sum_{i=1}^n c_i |{\psi_i}\rangle=0\;\text{then}\; c_i=0\;\text{for all}\; i$$
!!! info "" !!! info "Dimensionality"
The minimum number of vectors needed to form a complete set of basis states is known as the *dimensionality* of the state space. In quantum mechanis you will encounter systems whose Hilbert spaces have very different dimensionality, from the spin-1/2 particle (a $n=2$ vector space) to the free particle (whose state vectors live in an infinite vector space). The minimum number of vectors needed to form a complete set of basis states is known as the *dimensionality* of the state space. In quantum mechanis you will encounter systems whose Hilbert spaces have very different dimensionality, from the spin-1/2 particle (a $n=2$ vector space) to the free particle (whose state vectors live in an infinite vector space).
### Bra vectors ### Bra vectors
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