# Difference between revisions of "Quantum Mechanics"

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==Measurement== | ==Measurement== | ||

+ | For | ||

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+ | {| class="wikitable" | ||

+ | |- | ||

+ | ! Integral Notation | ||

+ | ! Dirac Notation | ||

+ | ! Matrix Notation | ||

+ | |- | ||

+ | | <math> \psi = \sum_n c_n \psi_n </math> | ||

+ | | <math> | ||

+ | \begin{alignat}{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{alignat} | ||

+ | </math> | ||

+ | | <math> |\psi\rangle = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}</math> | ||

+ | |- | ||

+ | | <math> \phi^* = \sum_n a_n^* \phi_n^* </math> | ||

+ | | | ||

+ | | <math> \langle \phi | = \begin{bmatrix} a_1^* & a_2^* & \dots & a_n^* \end{bmatrix} </math> | ||

+ | |||

+ | |} | ||

+ | |||

The transition of the wavefunction <math>\psi</math> into state <math>\phi</math> can be thought of as: | The transition of the wavefunction <math>\psi</math> into state <math>\phi</math> can be thought of as: | ||

{| class="wikitable" | {| class="wikitable" |

## Revision as of 14:56, 16 June 2021

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

## Contents

## Wavefunction

A quantum system is completely specified by its **Wave Function**:

Integral Notation | Dirac Notation |
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The wavefunction is typically normalized:

Integral Notation | Dirac Notation |
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The distribution of the particle described by is given by:

Integral Notation | Dirac Notation |
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In the Copenhagen Interpretation, is the probability of finding the particle at location . In Universal Wave Function interpretations (e.g. MWI), 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:

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

## State

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

A basis should be orthonormal:

Integral Notation | Dirac Notation | |
---|---|---|

normalized | ||

orthogonal |

An operator defines a particularly convenient basis: one can always expand using the basis defined by an operator, in which case the 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:

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

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

## Wave packet

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

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.)

## Superposition

If and are both allowed states for a given system, then the following state is also allowed:

where and are complex-valued coefficients. This leads to a notable consequence:

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 |
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## Operators

Observables in QM appears as operators ().

Examples:

Operator | Position basis | Momentum basis |
---|---|---|

Position | ||

Momentum | ||

Hamiltonian | ||

Energy |

## Commutators

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

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

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

Then the related uncertainties follow:

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:

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).

## Measurement

For

Integral Notation | Dirac Notation | Matrix Notation |
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The transition of the wavefunction into state can be thought of as:

Thus is the probability amplitude for the initial state to 'collapse' into the state . Measurements take the form of operators in QM. When acting on a wavefunction with operator the probability that the wavefunction ends up in state is given by:

The solutions take the form of an eigenvalue problem:

The allowed solutions of the equation, for operator , involve an eigenstate with associated eigenvalue . 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. , implies an expectation value (for state ) of:

If the system is in an eigenstate of the operator:

We know that:

And so:

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 () acts on the system state () to yield an energy eigenvalue ():

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

### 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:

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

## Entanglement

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

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

If all the coefficients can be written as , then there is no coupling between the two states. We call the system **separable**, since it can be decomposed into the two sub-systems:

However, if , then the states are non-separable, or **entangled**.

The entanglement of an observer (or simply measurement apparatus) can also be considered. For observer performing measurement on system :

This leads to the evolution:

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.

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:

For instance, a **pure state** is given by a density matrix that has only a single non-zero term, along the diagonal:

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

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**:

The off-diagonal terms indicate the strength and nature of the interference between the states. During decoherence, these off-diagonal terms are driven towards zero.

## Decoherence

TBD