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
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 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)} is given by:

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

In the Copenhagen Interpretation, 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)} is the probability of finding the particle at location 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 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:

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) \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 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} using the basis defined by an operator, in which case the 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_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:

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 \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.)

$\Delta _{x}\Delta _{p}\geq {\frac {\hbar }{2}}$

$\Delta _{E}\Delta _{t}\geq {\frac {\hbar }{2}}$

Superposition

If $\psi _{1}(x)$ and $\psi _{2}(x)$ are both allowed states for a given system, then the following state is also allowed:

$\psi (x)=\alpha \psi _{1}(x)+\beta \psi _{2}(x)$

This leads to a notable consequence:

${\begin{alignedat}{2}\Pr(x)&=|\alpha \psi _{1}(x)+\beta \psi _{2}(x)|^{2}\\&=(\alpha \psi _{1}+\beta \psi _{2})(\alpha \psi _{1}+\beta \psi _{2})^{*}\\&=|\alpha |^{2}|\psi _{1}|^{2}+|\beta |^{2}\psi _{2}^{2}+\alpha \beta ^{*}\psi _{1}\psi _{2}^{*}+\alpha ^{*}\beta \psi _{1}^{*}\psi _{2}\\&=\mathrm {Pr} _{1}(x)+\mathrm {Pr} _{2}(x)+\mathrm {interference} \\\end{alignedat}}$

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

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:

Failed to parse (unknown function "\begin{alignat}"): {\displaystyle \begin{alignat}{2} \psi = \sum_n c_n \psi_n = \psi_n }

We know that:

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

And so:

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} \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{alignat} }

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

See Also

• Wikipedia: Quantum Mechanics