Difference between revisions of "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
${\displaystyle \psi (x)}$   ${\displaystyle \langle x|\psi \rangle }$

The wavefunction is typically normalized:

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

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

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

In the Copenhagen Interpretation, ${\displaystyle \Pr(x)}$ is the probability of finding the particle at location ${\displaystyle x}$. In Universal Wave Function interpretations (e.g. MWI), ${\displaystyle \Pr(x)}$ 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:

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

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

${\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi }}}\int {\tilde {\psi }}(k)e^{ikx}\mathrm {d} k}$
${\displaystyle {\tilde {\psi }}(k)={\frac {1}{\sqrt {2\pi }}}\int {\psi }(x)e^{-ikx}\mathrm {d} x}$
 Example of a 1D wavefunction (plotted versus 'x'). The upper panel shows the amplitude-squared, while the lower panel shows the corresponding wavefunction components: real (black line) and imaginary (blue line).

State

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

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

A basis should be orthonormal:

Integral Notation Dirac Notation
${\displaystyle \int |\psi _{n}(x)|^{2}\mathrm {d} x=1}$   ${\displaystyle \langle \psi _{n}|\psi _{n}\rangle =1}$  normalized
${\displaystyle \int \psi _{m}(x)^{*}\psi _{n}(x)\mathrm {d} x=0}$   ${\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}$  orthogonal

An operator defines a particularly convenient basis: one can always expand ${\displaystyle \psi }$ using the basis defined by an operator, in which case the ${\displaystyle \psi _{n}}$ 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:

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

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

${\displaystyle \langle \psi |={\begin{bmatrix}c_{1}^{*}&c_{2}^{*}&\dots &c_{n}^{*}\end{bmatrix}}}$

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

${\displaystyle \langle \psi |^{\dagger }=|\psi \rangle }$

Wave packet

A wave packet is a localized wavelike perturbation. Particles in quantum mechanics can be thought of as wave-packets.

A 1D wave packet (with dispersion), propagating over time.

Note that "wave-particle duality" can be misleading. One can imagine a quantum particle as "both a wave and a particle"; however, it might be better to instead imagine it as a "wave packet". The 'particle' and 'wave' descriptions are really idealized limiting cases, which never appear in reality:

• A classical 'particle' is a point-like object. In QM would have a corresponding infinite spread in its momentum. Such an idealized (infinitely small) entity cannot truly exist.
• A classical 'wave' is a plane wave: an oscillation with a perfectly well-defined wavelength, extending infinitely in both directions. In QM, we indeed note that having a precisely-defined wavelength (momentum) implies infinite spatial spread (i.e. the wave fills the entire universe). Such a construct is not physically-realizable.

Heisenberg Indeterminacy Relations

(Also known as Heisenberg Uncertainty Principle.)

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

Superposition

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

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

where ${\displaystyle \scriptstyle \alpha }$ and ${\displaystyle \scriptstyle \beta }$ are complex-valued coefficients. This leads to a notable consequence:

{\displaystyle {\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 Matrix Notation
${\displaystyle \psi =\sum _{n}c_{n}\psi _{n}}$   {\displaystyle {\begin{alignedat}{2}|\psi \rangle &=c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle +c_{3}|\psi _{3}\rangle +\cdots \\&=\sum _{n}c_{n}|\psi _{n}\rangle \end{alignedat}}}   ${\displaystyle |\psi \rangle ={\begin{bmatrix}c_{1}\\c_{2}\\\vdots \\c_{n}\end{bmatrix}}}$

Operators

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

Examples:

Operator Position basis Momentum basis
Position  ${\displaystyle {\hat {x}}=x}$   ${\displaystyle {\hat {x}}=i\hbar {\frac {\partial }{\partial p}}}$
Momentum  ${\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}$   ${\displaystyle {\hat {p}}=p}$
Hamiltonian  ${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)}$
Energy  ${\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}$

Commutators

The commutator is a useful function for evaluating quantum objects. It is defined by:

${\displaystyle \left[{\hat {A}},{\hat {B}}\right]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}}$

The commutator can be thought of as a composite operator, which can thus act on a wavefunction:

${\displaystyle \left[{\hat {A}},{\hat {B}}\right]\psi ={\hat {A}}{\hat {B}}\psi -{\hat {B}}{\hat {A}}\psi }$

The uncertainty relations are intimately tied to commutators of operators. For instance, for two non-commuting operators:

${\displaystyle [{\hat {A}},\,{\hat {B}}]=c\mathbf {1} }$

Then the related uncertainties follow:

${\displaystyle \Delta _{\psi }A\Delta _{\psi }B\geq {\frac {1}{2}}|\langle \psi |[{\hat {A}},\,{\hat {B}}]|\psi \rangle |}$

Therefore, non-commuting operators have a indeterminacy relation between their associated observables; and the commutation relation provides the bound on this inter-relation. The two observables cannot both be simultaneously definite (the wavefunction cannot be in a state that is simultaneously an eigenstate of both operators, so it cannot have a definite eigenvalue for both of the observables). On the other hand, if two operators commute:

${\displaystyle [{\hat {A}},\,{\hat {B}}]=\mathbf {0} }$

Then the wavefunction can be in a state that is simultaneously an eigenstate of both operators. There is thus no restriction on both observables (A and B) being simultaneously definite (known to arbitrary precision).

${\displaystyle \Delta _{\psi }A\Delta _{\psi }B\geq 0}$

Measurement

For state ${\displaystyle \psi }$ and linear map ${\displaystyle \langle \phi |}$:

Integral Notation Dirac Notation Matrix Notation
${\displaystyle \psi \in V}$
${\displaystyle \psi =\sum _{n}c_{n}\psi _{n}}$
{\displaystyle {\begin{alignedat}{2}|\psi \rangle &=c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle +c_{3}|\psi _{3}\rangle +\cdots \\&=\sum _{n}c_{n}|\psi _{n}\rangle \end{alignedat}}}   ${\displaystyle |\psi \rangle ={\begin{bmatrix}c_{1}\\c_{2}\\\vdots \\c_{n}\end{bmatrix}}}$
${\displaystyle f_{\phi }\in V^{*}}$   ${\displaystyle \langle \phi |=a_{1}^{*}\langle \phi _{1}|+a_{2}^{*}\langle \phi _{2}|+a_{3}^{*}\langle \phi _{3}|+\dots }$   ${\displaystyle \langle \phi |={\begin{bmatrix}a_{1}^{*}&a_{2}^{*}&\dots &a_{n}^{*}\end{bmatrix}}}$

The linear map ${\displaystyle \langle \phi |}$ is a function (in the complex vector space ${\displaystyle V^{*}}$, dual to ${\displaystyle V}$) that maps from vector space ${\displaystyle V}$ to ${\displaystyle \mathbb {C} }$. In conventional notation this operation would be performed using a function ${\displaystyle f_{\phi }}$ applied to ${\displaystyle \psi }$:

${\displaystyle f_{\phi }(\psi )=a_{1}^{*}\psi _{1}+a_{2}^{*}\psi _{2}+\dots +a_{n}^{*}\psi _{n}}$

In a quantum measurement, the transition of the wavefunction ${\displaystyle \psi }$ into state ${\displaystyle \phi }$ can be thought of as:

 ${\displaystyle \int \phi ^{*}\psi \mathrm {d} x}$ ${\displaystyle \langle \phi |\psi \rangle =a_{1}^{*}c_{1}+a_{2}^{*}c_{2}+a_{3}^{*}c_{3}+\cdots }$

Thus ${\displaystyle \scriptstyle \langle \phi |\psi \rangle }$ is the probability amplitude for the initial state ${\displaystyle \scriptstyle \psi }$ to 'collapse' into the state ${\displaystyle \scriptstyle \phi }$. Measurements take the form of operators in QM. When acting on a wavefunction with operator ${\displaystyle {\hat {O}}}$ the probability that the wavefunction ends up in state ${\displaystyle \phi _{n}}$ is given by:

 {\displaystyle {\begin{alignedat}{2}\Pr(O_{n})&=\left|\int \phi _{n}^{*}\psi \mathrm {d} x\right|^{2}\\&=\left|\int \phi _{n}^{*}\sum _{m}c_{m}\phi _{m}\mathrm {d} x\right|^{2}\\&=\left|c_{n}\int \phi _{n}^{*}\phi _{n}\mathrm {d} x\right|^{2}\\&=|c_{n}|^{2}\end{alignedat}}} {\displaystyle {\begin{alignedat}{2}\Pr(O_{n})&=|\langle n|\psi \rangle |^{2}\\&=|c_{n}|^{2}\end{alignedat}}}

The solutions take the form of an eigenvalue problem:

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

The allowed solutions of the equation, for operator ${\displaystyle {\hat {O}}}$, involve an eigenstate ${\displaystyle \phi _{n}}$ with associated eigenvalue ${\displaystyle o_{n}}$. 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. ${\displaystyle {\hat {A}}}$, implies an expectation value (for state ${\displaystyle \psi }$) of:

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

If the system is in an eigenstate of the operator:

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

We know that:

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

And so:

{\displaystyle {\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

Time-independent equation

This simplified version of the Schrödinger equation can be used to solve for allowed stationary states. The general form is akin to the eigenvalue problems noted above: the energy operator (${\displaystyle {\hat {H}}}$) acts on the system state (${\displaystyle \Psi }$) to yield an energy eigenvalue (${\displaystyle E}$):

${\displaystyle E\Psi ={\hat {H}}\Psi }$

For a single non-relativistic particle, the Hamiltonian is known and the Schrödinger equation takes the form:

${\displaystyle E\Psi (\mathbf {r} )=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right]\Psi (\mathbf {r} )}$

Time-dependent equation

More generally, the time-evolution of the wavefunction should be considered. The full version of the Schrödinger equation thus includes time dependence:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$

Again for a single non-relativistic particle, we can write more specifically that:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[{\frac {-\hbar ^{2}}{2\mu }}\nabla ^{2}+V(\mathbf {r} ,t)\right]\Psi (\mathbf {r} ,t)}$

Entanglement

When systems (or Hilbert spaces) A and B interact, they become entangled. Before the interaction, the two systems are simply a composite system:

${\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}}$

At this level, states are separable. However, the composite system more generally should be written as:

${\displaystyle |\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}}$

If all the coefficients can be written as ${\displaystyle \scriptstyle c_{ij}=c_{i}^{A}c_{j}^{B},}$, then there is no coupling between the two states. We call the system separable, since it can be decomposed into the two sub-systems:

${\displaystyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}}$
${\displaystyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}}$

However, if ${\displaystyle \scriptstyle c_{ij}\neq c_{i}^{A}c_{j}^{B}}$, then the states are non-separable, or entangled.

The entanglement of an observer (or simply measurement apparatus) can also be considered. For observer ${\displaystyle \scriptstyle |\psi \rangle }$ performing measurement ${\displaystyle \scriptstyle U}$ on system ${\displaystyle \scriptstyle \sum a_{n}|S_{n}\rangle }$:

${\displaystyle U(|\psi \rangle \otimes |S_{n}\rangle )=|\psi _{n}\rangle \otimes |S_{n}\rangle }$

This leads to the evolution:

${\displaystyle |\psi \rangle \otimes \sum a_{n}|S_{n}\rangle \to \sum a_{n}|\psi _{n}\rangle \otimes |S_{n}\rangle }$

Where the right-hand-side describes an entanglement between the observer and the system being studied.

Density Matrices

The outer product of a ket with a bra defines a 2D matrix; i.e. a linear operator.

${\displaystyle |\phi \rangle \langle \psi |={\begin{bmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{bmatrix}}{\begin{bmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{bmatrix}}={\begin{bmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{bmatrix}}}$

Density matrices can be a useful way to visualize the interactions between states of a system, and time-evolution. Thus we can define the density matrix as:

${\displaystyle \rho (t)\equiv |\psi (t)\rangle \langle \psi (t)|}$

The elements of the density matrix can be interpreted as a form of quantum probability distribution. For instance, a pure state is given by a density matrix that has only a single non-zero term, along the diagonal:

${\displaystyle |\psi \rangle \langle \psi |={\begin{bmatrix}1&0\\0&0\end{bmatrix}}}$

By comparison, a mixed state is one where more than one diagonal term is non-zero, but off-diagonal terms are zero:

${\displaystyle |\psi \rangle \langle \psi |={\begin{bmatrix}0.5&0\\0&0.5\end{bmatrix}}}$

The diagonal terms of the density matrix will be real numbers (and can be interpreted as probabilities). The above matrix describes a classical mixture of different states. In the Copenhagen interpretation, this would be described as a situation where the system will randomly collapse into one of the possible states. Under MWI, it would instead be said that the various states are now non-interacting, and thus will evolve independently.

The most general case is where the off-diagonal terms are non-zero. These terms represent the interference aspects. Thus this describes an entangled state:

${\displaystyle |\psi \rangle \langle \psi |={\begin{bmatrix}0.36&-0.48i\\+0.48i&0.64\end{bmatrix}}}$

The off-diagonal terms indicate the strength and nature of the interference between the states. Thus, the diagonal elements (${\displaystyle n=m}$) capture the probability of occupying a certain state as ${\displaystyle \rho _{nm}={\overline {c_{n}c_{n}^{*}}}=p_{n}\geq 0}$The time-dependence often appears in these off-diagonal (${\displaystyle n\neq m}$) terms as ${\displaystyle \rho _{nm}(t)={\overline {c_{n}(t)c_{m}^{*}(t)}}={\overline {c_{n}c_{m}^{*}}}e^{-i\omega _{nm}t}}$.

During decoherence, these off-diagonal terms are driven towards zero.

TBD