Wave function
The wave function in quantum mechanics describing the quantum state of an elementary particle or a system of elementary particles in the spatial domain, their absolute square determines the probability of finding the particle. According to the Copenhagen interpretation of quantum mechanics, the wave function contains a description of all the information an entity or an entire system.
A wave function is the function that solves the Schrödinger equation ( in real space ). Solutions of these equations can shaft (such as a α - or β - particles as the wave packet ) describe both bound particles (such as electrons in the shell of an atom ) or free particles. The wave function is a complex function in the rule.
For particle systems (eg with several identical particles ) refers to such a solution as a many-body wave function. These solutions can be, due to the interaction of the particles with each other, but usually not calculate without the more modern methods of quantum field theory.
- 3.1 A particle in one spatial dimension
- 3.2 A particle in three spatial dimensions
- 3.3 Two distinguishable particles in three space dimensions
- 3.4 A particle in one-dimensional momentum space
- 3.5 Spin 1/2- particles (e.g. electron)
- 4.1 Finite vectors
- 4.2 Infinite vectors
- 4.3 Steady -indexed vectors ( functions)
Quantum particle as a wave
In the Schrödinger between quantum mechanics wave functions arise as a solution of the Schrödinger equation. So, if one uses a function in the Schrödinger equation, they must solve the equation to be a wave function of the Schrödinger equation.
The fact that the wave function is not real, but complex-valued function is reflected inter alia, that does not necessarily play a real physical meaning. It is, as a rule can not be measured and is only the mathematical description of the quantum mechanical state of a physical system. From this, however, the expected measurement by complex conjugation can be calculated. That the wave function is complex, followed in turn of the Schrödinger equation. This is defined in the complex space. Therefore, it needs the general solution is usually a function whose function values in the complex space lie.
For comparison, the electric field strength of a radio wave is the solution of a classical electrodynamic wave equation. However, this electric field strength can be measured, for example, by an antenna and a radio receiver.
Particles having intrinsic properties ( such as the spun-bonded electron or the angular momentum of a photon ) can be described by the wave functions with a plurality of components. Depending on the transformation behavior of the wave functions under Lorentz transformations, a distinction is scalar, tensor and spinorielle wave functions or fields in relativistic quantum field theory.
Normalization condition and probability
In contrast to classical physics an exact statement about the whereabouts of a particle is generally not possible ( Heisenberg uncertainty principle ). Starting from the existence of the quantum particle, it must be (at any time ) to stop somewhere, so the wave function of the spatial normalization condition
Must meet (the complex conjugate function ).
This leads to the differential probability of finding the particle at the position in the volume element.
For a normalized wave function of the absolute square there is the probability density for the stay at the place at the time t. Particle wave functions in the spatial domain, the integration of the probability density is over a local area (an interval in space) (e.g., electron) to the probability of finding a particle at this spatial region.
Simple wave function
Wave function
The wave function of a quantum free particle can have for example the shape of a plane wave having a ( mathematically ) real and a ( mathematically ) imaginary part:
In which
- The place as a vector ( position vector )
- The ( complex-valued ) amplitude, yielding the real part and the imaginary, so that
- The wave vector, which determines the direction and wavelength of the wave, and
- Is the angular frequency, which describes the oscillation period of the shaft.
Measurement
The wave function multiplied by your complex conjugation yields the absolute square of the wave function. This is a function that specifies the differential probability of finding the particle as a function of position and time:
However, the absolute square of the wave function does not describe directly in this case to meet the Wahrscheinlichtkeit a particle in position space, since the function is not standardized.
Standardization
The probability must be over the spatial domain integrated 1 result. Thus, it can be interpreted that, in the entire space, a particle can be found exactly. To this end, we can calculate the integral over and shares the wave function by the square root of this integral to normalize this. Of the normalized wave function is obtained so that the correct probability density function.
However, the integral over a plane wave is not defined.
For this reason, the wave function multiplied by an envelope function ( e.g. a gauss function). The resulting function can have a predictable finite integral. In addition, they can have practically the same properties for all other applications such as. In this case one speaks of a wave packet.
Definition
A wave function refers to any vector or any function that describes the state of a physical system, by representing it as a development for other states of the same system.
Typical wave functions are either:
- A vector of complex numbers with a finite number of components:
- A vector of complex numbers with countably infinitely many components ( discrete index):
- Or a complex-valued function of one or more continuously variable real variables:
In all cases, the wave function provides a full description of the physical system in question. However, it is important to note that a particular system associated wave function, the system is not uniquely determined, but rather many different wave functions to describe the same physical scenario.
Particle interpretation
The physical interpretation of a wave function is context dependent. Several examples are provided below, followed by an interpretation of the three cases described above.
A particle in one spatial dimension
The wave function of a particle in one-dimensional space is a complex function over the set of real numbers. The sum of the square wave function is interpreted as a probability density of the particle position.
The probability of finding a particle in a measurement in the interval, thus
This leads to the normalizing condition
As a measurement of the particle position must result in a real number. This means that the probability of finding a particle at any location is equal to 1
A particle in three dimensions
The three-dimensional case is analogous to the one-dimensional; The wave function is a complex function defined by the three-dimensional space, and its amount is interpreted as a three-dimensional square probability density. The probability of finding a particle in a measurement volume, is thus
The normalization condition is analogous to the one-dimensional case
Where the integral extends over the entire room.
Two distinguishable particles in three space dimensions
In this case, the wave function is a complex function of six spatial variables,
And the joint probability density function of the positions of the two particles. The probability of a position measurement of both the particles in the two respective regions R and S is then
Being and similarly for. The normalization condition is therefore
Wherein the pre- set integral ranges over the entire region of all six variables.
It is crucial that in case of two-particle systems, only the system, which consists of two particles must have a well-defined wave functions. It follows that it may be impossible to define a probability density for particles ONE, which are not explicitly dependent on the position of particles of two. The Modern physics calls this phenomenon quantum entanglement and quantum non-locality.
A particle in the one-dimensional momentum space
The wave function of a one-dimensional particle in momentum space is a complex function defined on the set of real numbers. The size is interpreted as a probability density function in momentum space. The probability of a pulse measurement gives a value in the interval, thus
This leads to the normalizing condition
Because a measurement of the particle always yields a real number.
Spin 1/2- particles (e.g. electron)
The wave function of a particle with spin 1/2 (without consideration of its spatial degrees of freedom) is a column vector
The importance of the components of the vector depends on the base used from, and typically correspond to the coefficients for an alignment of the spins in direction ( spin up ) and opposite to the direction ( spin down ). In the Dirac notation this is:
And the values are then interpreted as the probability that the pin is oriented in a measurement in the direction of or opposite to the direction.
This leads to the normalizing condition
Basic interpretation of the vector representation
Wave functions can be used as elements of a vector space conceived ( Hilbert space ). A wave function that describes the state of a physical system can be described by a linear combination of the other states of the same system. We call the state of the system under consideration and as the states in which it is designed, as well. The latter states should constitute a basis of the vector space. In the following, all wave functions are assumed to be normalized.
Finite vectors
A vectorial wave function with components describes how as a linear combination of finitely many basic elements, which run from expressing the condition of the physical system. In particular, the equation
Which is a relation between column vectors, equivalent to the basic decomposition
Which is a relation between the states of a physical system. It should be noted that one must know the basis used when switching between these expressions, and consequently two column vectors with the same components represent two different system states when the underlying basis states are different. An example of a finite wave vector function is given by the spin state of a particle with spin 1 /2, as described above.
The physical meaning of the components of is given by the postulate of the collapse of the wave function:
( Contrast, momentum and position variables have a continuous spectrum. Them In the base decomposition is given by an integral, that is, the above sum representation for is to be replaced by an integral, and the probabilities by expressions of the form. )
Infinite vectors
The case of infinite vectors with discrete index is treated the same as a finite vector, except that the sum over all ( infinitely many ) basic elements is extended. Consequently,
Equivalent to
Where in the above sum of all components are taken into account by. The interpretation of the components is the same as in the finite case ( the measurement is interpreted just as for vectors from a finite-dimensional Hilbert space: compare the wave function collapse ).
Continuous indexed vectors ( functions)
If the index is not discrete but continuous, the sum is replaced by an integral; an example of this is the local view of the wave function of a particle in one dimension, which is the ( abstract ) state of the particle in a particular local base:
Here, the state vector is not to be confused with its " component representation " in the spatial domain. The former expression indicates the condition of the particle, abstract and without reference to a specific basis representation, while the latter refers to the wave function in the spatial domain, which is interpreted as a superposition of basis states with defined positions. The basis states can also act as integral
Be formulated. Thus, a belonging to the wave function is written in the spatial domain as a delta function. It should be noted that the latter is, however, not as an ordinary wave function in the Hilbert space of square-integrable functions.
Formalism
Given an isolated physical system, the allowed states are (ie, the states can adopt the system without violating the laws of physics ) a subset of a vector space H, the Hilbert space. Specifically, this subset is the set of all vectors of length 1, ie the unit ball around the origin. This follows from the fact that all states are physically allowed normalized. It follows:
- If and two allowed states, then is also a permitted state if and only if the following applies (normalization ).
- Because of the normalization can be found from physically allowed states for the vector space H is always an orthonormal basis.
In this context, the wave function of a particular condition can be regarded as a development of a condition on a basis of the vector space. For example,
A base of the space, which describes a particle with spin 1/2, and it follows that the spin state of a particle such by
It is common to be provided with an inner product ( dot product ), the nature of the inner product is dependent on the base used. If there is a countable set of basis elements which belong to all, are, then with the unique inner product that makes this orthonormal basis, provided, eg:
When that is done, the inner product of the development of an arbitrary vector is
The coefficient of the development of the state into the base so obtained by projection onto the basis vector.
If the eigenvalues form a continuum, which is the case for example in the local or coordinate - based, not a Hilbert space can be constructed from the corresponding eigenstates, since these eigenstates are not square integrable. By using the Dirac delta function can, however, formulate a generalized orthonormalization condition for these basis states. Such bases are also referred to as improper bases. One example is the above-mentioned development of the spatial wave function of a particle in states of definite position, with the Dirac normalization
So that the analogous identity
Is satisfied.