Quantum Mechanics

Quantum mechanics is a theory that describes the interactions of all particles and systems. It underlies all physical phenomena, including scattering.

Wavefunction

A quantum system is completely specified by its Wave Function:

Integral Notation	Dirac Notation
$\psi (x)$	$\langle x|\psi \rangle$

The wavefunction is typically normalized:

Integral Notation	Dirac Notation
$\int |\psi (x)|^{2}\mathrm {d} x=1$	$\langle \psi |\psi \rangle =1$

The distribution of the particle described by $\psi (x)$ is given by:

Integral Notation	Dirac Notation
$\Pr(x)\mathrm {d} x=|\psi (x)|^{2}\mathrm {d} x$	$|\langle x|\psi \rangle |^{2}$

In the Copenhagen Interpretation, $\Pr(x)$ is the probability of finding the particle at location $x$. In Universal Wave Function interpretations (e.g. MWI), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pr(x)} can be thought of as the spatial distribution of the particle. The wavefunction contains all the information one can know about a system. It can thus be thought of as 'being' the particle/system in question. However, the wavefunction can be described in an infinite number of different ways. That is, there is not a unique basis for describing the wavefunction. So, for instance, one can describe the wavefunction using position-space or momentum-space:

$\psi (x)\longleftrightarrow {\tilde {\psi }}(k)$

These representations can be inter-related (c.f. Fourier transform):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x) = \frac{1}{\sqrt{2 \pi}} \int \tilde{\psi}(k) e^{i k x } \mathrm{d}k }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\psi}(k) = \frac{1}{\sqrt{2 \pi}} \int {\psi}(x) e^{-i k x } \mathrm{d}x }

State

Note that the wavefunction describes the state of the system; there are various choices of basis one can use as an expansion.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = \sum_n c_n \psi_n }

An operator defines a particular convenient basis: one can always expand $\psi$ using the basis defined by an operator, in which case the $\psi _{n}$ above are the eigenvectors (or eigenstates) of that basis. This can also be viewed as a vector in the Hilbert space. The Dirac notation (bra-ket notation) is useful in this regard. A particular state is a (column) vector:

$|\psi \rangle ={\begin{bmatrix}c_{1}\\c_{2}\\\vdots \\c_{n}\end{bmatrix}}$

Which is a 'ket'. We define a 'bra' (the 'final state') as a (row) vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi | = \begin{bmatrix} c_1^* & c_2^* & \dots & c_n^* \end{bmatrix} }

And note that the 'bra' is the conjuagte transpose of the 'ket':

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi | ^{\dagger} = | \psi \rangle }

Wave packet

TBD

Heisenberg Indeterminacy Relations

(Also known as Heisenberg Uncertainty Principle.)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_{x}\Delta_{p} \geq \frac{\hbar}{2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_{E}\Delta_{t} \geq \frac{\hbar}{2}}

Superposition

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_1(x)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_2(x)} are both allowed states for a given system, then the following state is also allowed:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x) = \alpha \psi_1(x) + \beta \psi_2(x) }

This leads to a notable consequence:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \Pr(x) & = | \alpha \psi_1(x) + \beta \psi_2(x) |^2 \\ & = (\alpha\psi_1 + \beta\psi_2)(\alpha\psi_1 + \beta\psi_2)^{*} \\ & = |\alpha|^2 |\psi_1|^2 + |\beta|^2\psi_2^2 + \alpha\beta^* \psi_1\psi_2^* + \alpha^*\beta\psi_1^*\psi_2 \\ & = \mathrm{Pr}_1(x) + \mathrm{Pr}_2(x) + \mathrm{interference} \\ \end{alignat} }

Notice that the final terms represent 'interference' between the two constituent states. This interference has no classical analogue; it is a quantum effect. Thus a superposition is not merely a 'joining' of the two states (e.g. "the particle can be in state 1 or state 2"), but a truly coherent interference between the two states. The superposition may be more generally written as:

Integral Notation	Dirac Notation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int | \psi(x) |^2 \mathrm{d}x = 1}	$\langle \psi |\psi \rangle =1$

The distribution of the particle described by $\psi (x)$ is given by:

Integral Notation	Dirac Notation
$\psi (x)=\sum _{n}c_{n}\psi _{n}$	$|\psi \rangle =c_{1}|1\rangle +c_{2}|2\rangle +c_{3}|3\rangle +\cdots$

Operators

Observables in QM appears as operators (${\hat {O}}$).

Examples: TBD.

Measurement

The transition of the wavefunction $\psi$ into state $\phi$ can be thought of as:

$\int \phi ^{*}\psi \mathrm {d} x$

$\langle \phi |\psi \rangle =a_{1}^{*}c_{1}+a_{2}^{*}c_{2}+a_{3}^{*}c_{3}+\cdots$

When acting on a wavefunction with operator ${\hat {O}}$ the probability that the wavefunction ends up in state $\phi _{n}$ is given by:

$\Pr(O_{n})=|c_{n}|^{2}$

$\Pr(O_{n})=|\langle n|\psi \rangle |^{2}=|c_{n}|^{2}$

The solutions take the form of an eigenvalue problem:

${\hat {O}}\phi _{n}=o_{n}\phi _{n}$

The allowed solutions of the equation, for operator ${\hat {O}}$, involve an eigenstate $\phi _{n}$ with associated eigenvalue $o_{n}$. A measurement on a quantum system can be thought of as driving the wavefunction into an eigenstate defined by the operator; the value of the associated observable is then fixed to be the corresponding eigenvalue. (As noted above, the probability of ending up in a particular eigenstate is regulated by the coefficient of that eigenstate in the original wavefunction decomposition.)

Expectation value

A given operator, e.g. ${\hat {A}}$, implies an expectation value of:

$\langle A\rangle _{\psi }=\int \psi ^{*}{\hat {A}}\psi \mathrm {d} x$

$\langle A\rangle _{\psi }=\langle \psi |{\hat {A}}|\psi \rangle$

If the system is in an eigenstate of the operator:

$\psi =\sum _{n}c_{n}\psi _{n}=\psi _{n}$

We know that:

${\hat {A}}\psi _{n}=a_{n}\psi _{n}$

And so:

${\begin{alignedat}{2}\langle A\rangle &=\int \psi _{n}^{*}{\hat {A}}\psi _{n}\mathrm {d} x\\&=\int \psi ^{*}a_{n}\psi \mathrm {d} x\\&=a_{n}\int \psi ^{*}\psi \mathrm {d} x\\&=a_{n}\\\end{alignedat}}$

In other words, the expectation value of an eigenstate is simply the eigenvalue.

Schrödinger Equation

TBD

Entanglement

TBD

Decoherence

TBD

See Also

• Wikipedia: Quantum Mechanics