Wave function, or psi function ψ (\displaystyle \psi )- a complex-valued function used in quantum mechanics to describe the pure state of a system. Is the coefficient of expansion of the state vector over a basis (usually a coordinate one):
| ψ (t) ⟩ = ∫ Ψ (x, t) | x ⟩ d x (\displaystyle \left|\psi (t)\right\rangle =\int \Psi (x,t)\left|x\right\rangle dx)
Where | x⟩ = | x 1 , x 2 , … , x n ⟩ (\displaystyle \left|x\right\rangle =\left|x_(1),x_(2),\ldots ,x_(n)\right\rangle ) is the coordinate basis vector, and Ψ(x, t) = ⟨x | ψ (t) ⟩ (\displaystyle \Psi (x,t)=\langle x\left|\psi (t)\right\rangle )- wave function in coordinate representation.
Normalization of the wave function
Wave function Ψ (\displaystyle \Psi ) in its meaning must satisfy the so-called normalization condition, for example, in the coordinate representation having the form:
∫ V Ψ ∗ Ψ d V = 1 (\displaystyle (\int \limits _(V)(\Psi ^(\ast )\Psi )dV)=1)This condition expresses the fact that the probability of finding a particle with a given wave function anywhere in space is equal to one. In the general case, integration must be carried out over all variables on which the wave function in a given representation depends.
Principle of superposition of quantum states
For wave functions, the principle of superposition is valid, which consists in the fact that if a system can be in states described by wave functions Ψ 1 (\displaystyle \Psi _(1)) And Ψ 2 (\displaystyle \Psi _(2)), then it can also be in a state described by the wave function
Ψ Σ = c 1 Ψ 1 + c 2 Ψ 2 (\displaystyle \Psi _(\Sigma )=c_(1)\Psi _(1)+c_(2)\Psi _(2)) for any complex c 1 (\displaystyle c_(1)) And c 2 (\displaystyle c_(2)).
Obviously, we can talk about the superposition (addition) of any number of quantum states, that is, about the existence of a quantum state of the system, which is described by the wave function Ψ Σ = c 1 Ψ 1 + c 2 Ψ 2 + … + c N Ψ N = ∑ n = 1 N c n Ψ n (\displaystyle \Psi _(\Sigma )=c_(1)\Psi _(1)+ c_(2)\Psi _(2)+\ldots +(c)_(N)(\Psi )_(N)=\sum _(n=1)^(N)(c)_(n)( \Psi )_(n)).
In this state, the square of the modulus of the coefficient c n (\displaystyle (c)_(n)) determines the probability that, when measured, the system will be detected in a state described by the wave function Ψ n (\displaystyle (\Psi )_(n)).
Therefore, for normalized wave functions ∑ n = 1 N | c n | 2 = 1 (\displaystyle \sum _(n=1)^(N)\left|c_(n)\right|^(2)=1).
Conditions for the regularity of the wave function
The probabilistic meaning of the wave function imposes certain restrictions, or conditions, on wave functions in problems of quantum mechanics. These standard conditions are often called conditions for the regularity of the wave function.
Wave function in various representations states are used in different representations - will correspond to the expression of the same vector in different coordinate systems. Other operations with wave functions will also have analogues in the language of vectors. In wave mechanics, a representation is used where the arguments of the psi function are the complete system continuous commuting observables, and the matrix representation uses a representation where the arguments of the psi function are the complete system discrete commuting observables. Therefore, the functional (wave) and matrix formulations are obviously mathematically equivalent.
WAVE FUNCTION, in QUANTUM MECHANICS, a function that allows you to find the probability that a quantum system is in some state s at time t. Usually written: (s) or (s, t). The wave function is used in the SCHRÖDINGER equation... Scientific and technical encyclopedic dictionary
WAVE FUNCTION Modern encyclopedia
Wave function- WAVE FUNCTION, in quantum mechanics the main quantity (in the general case complex) that describes the state of a system and allows one to find the probabilities and average values of physical quantities characterizing this system. Wave module square... ... Illustrated Encyclopedic Dictionary
WAVE FUNCTION- (state vector) in quantum mechanics is the main quantity that describes the state of a system and allows one to find the probabilities and average values of physical quantities characterizing it. The squared modulus of the wave function is equal to the probability of a given... ... Big Encyclopedic Dictionary
WAVE FUNCTION- in quantum mechanics (probability amplitude, state vector), a quantity that completely describes the state of a micro-object (electron, proton, atom, molecule) and any quantum in general. systems. Description of the state of a microobject using V.f. It has… … Physical encyclopedia
wave function- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information technology in general EN wave function ... Technical Translator's Guide
wave function- (probability amplitude, state vector), in quantum mechanics the main quantity that describes the state of a system and allows one to find the probabilities and average values of physical quantities characterizing it. The squared modulus of the wave function is... ... encyclopedic Dictionary
wave function- banginė funkcija statusas T sritis fizika atitikmenys: engl. wave function vok. Wellenfunktion, f rus. wave function, f; wave function, f pranc. fonction d’onde, f … Fizikos terminų žodynas
wave function- banginė funkcija statusas T sritis chemija apibrėžtis Dydis, apibūdinantis mikrodalelių ar jų sistemų fizikinę būseną. atitikmenys: engl. wave function rus. wave function... Chemijos terminų aiškinamasis žodynas
WAVE FUNCTION- a complex function that describes the state of quantum mechanics. system and allows you to find probabilities and cf. the meanings of the physical characteristics it characterizes. quantities Square modulus V. f. is equal to the probability of a given state, therefore V.f. called also amplitude... ... Natural science. encyclopedic Dictionary
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Based on the idea that an electron has wave properties. Schrödinger in 1925 suggested that the state of an electron moving in an atom should be described by the equation of a standing electromagnetic wave, known in physics. Substituting its value from the de Broglie equation instead of the wavelength into this equation, he obtained a new equation relating the electron energy to spatial coordinates and the so-called wave function, corresponding in this equation to the amplitude of the three-dimensional wave process.
The wave function is especially important for characterizing the state of the electron. Like the amplitude of any wave process, it can take on both positive and negative values. However, the value is always positive. Moreover, it has a remarkable property: the greater the value in a given region of space, the higher the probability that the electron will manifest its action here, that is, that its existence will be detected in some physical process.
The following statement will be more accurate: the probability of detecting an electron in a certain small volume is expressed by the product . Thus, the value itself expresses the probability density of finding an electron in the corresponding region of space.
Rice. 5. Electron cloud of the hydrogen atom.
To understand the physical meaning of the squared wave function, consider Fig. 5, which depicts a certain volume near the nucleus of a hydrogen atom. The density of points in Fig. 5 is proportional to the value in the corresponding place: the larger the value, the denser the points are located. If an electron had the properties of a material point, then Fig. 5 could be obtained by repeatedly observing the hydrogen atom and each time marking the location of the electron: the density of points in the figure would be greater, the more often the electron is detected in the corresponding region of space or, in other words, the greater the probability of its detection in this region.
We know, however, that the idea of an electron as a material point does not correspond to its true physical nature. Therefore Fig. It is more correct to consider 5 as a schematic representation of an electron “smeared” throughout the entire volume of an atom in the form of a so-called electron cloud: the denser the points are located in one place or another, the greater the density of the electron cloud. In other words, the density of the electron cloud is proportional to the square of the wave function.
The idea of the state of an electron as a certain cloud of electric charge turns out to be very convenient; it conveys well the main features of the behavior of the electron in atoms and molecules and will be often used in the subsequent presentation. At the same time, however, it should be borne in mind that the electron cloud does not have specific, sharply defined boundaries: even at a great distance from the nucleus, there is some, albeit very small, probability of detecting an electron. Therefore, by electron cloud we will conventionally understand the region of space near the nucleus of an atom in which the predominant part (for example, ) of the charge and mass of the electron is concentrated. A more precise definition of this region of space is given on page 75.
corpuscular - wave dualism in quantum physics, the state of a particle is described using the wave function ($\psi (\overrightarrow(r),t)$- psi-function).
Definition 1
Wave function is a function that is used in quantum mechanics. It describes the state of a system that has dimensions in space. It is a state vector.
This function is complex and formally has wave properties. The movement of any particle of the microworld is determined by probabilistic laws. The probability distribution is revealed when a large number of observations (measurements) or a large number of particles are carried out. The resulting distribution is similar to the wave intensity distribution. That is, in places with maximum intensity, the maximum number of particles is noted.
The set of arguments of the wave function determines its representation. Thus, a coordinate representation is possible: $\psi(\overrightarrow(r),t)$, an impulse representation: $\psi"(\overrightarrow(p),t)$, etc.
In quantum physics, the goal is not to accurately predict an event, but to estimate the probability of a particular event. Knowing the probability value, find the average values of physical quantities. The wave function allows you to find such probabilities.
Thus, the probability of the presence of a microparticle in volume dV at time t can be defined as:
where $\psi^*$ is the complex conjugate function to the function $\psi.$ The probability density (probability per unit volume) is equal to:
Probability is a quantity that can be observed in an experiment. At the same time, the wave function is not available for observation, since it is complex (in classical physics, the parameters that characterize the state of a particle are available for observation).
Normalization condition for the $\psi$-function
The wave function is determined up to an arbitrary constant factor. This fact does not affect the state of the particle that the $\psi$-function describes. However, the wave function is chosen in such a way that it satisfies the normalization condition:
where the integral is taken over the entire space or over a region in which the wave function is not zero. Normalization condition (2) means that in the entire region where $\psi\ne 0$ the particle is reliably present. A wave function that obeys the normalization condition is called normalized. If $(\left|\psi\right|)^2=0$, then this condition means that there is probably no particle in the region under study.
Normalization of the form (2) is possible with a discrete spectrum of eigenvalues.
The normalization condition may not be feasible. So, if $\psi$ is a plane de Broglie wave and the probability of finding a particle is the same for all points in space. These cases are considered as an ideal model in which the particle is present in a large but limited region of space.
Wave function superposition principle
This principle is one of the main postulates of quantum theory. Its meaning is as follows: if for some system states are possible that are described by the wave functions $\psi_1\ (\rm and)\ $ $\psi_2$, then for this system there is a state:
where $C_(1\ )and\ C_2$ are constant coefficients. The principle of superposition is confirmed empirically.
We can talk about the addition of any number of quantum states:
where $(\left|C_n\right|)^2$ is the probability that the system is found in a state that is described by the wave function $\psi_n.$ For wave functions subject to the normalization condition (2), the following condition is satisfied:
Stationary states
In quantum theory, stationary states (states in which all observable physical parameters do not change over time) play a special role. (The wave function itself is fundamentally unobservable.) In a steady state, the $\psi$-function has the form:
where $\omega =\frac(E)(\hbar )$, $\psi\left(\overrightarrow(r)\right)$ does not depend on time, $E$ is the particle energy. With the form (3) of the wave function, the probability density ($P$) is a time constant:
From physical properties stationary states followed by the mathematical requirements for the wave function $\psi\left(\overrightarrow(r)\right)\to \ (\psi(x,y,z))$.
Mathematical requirements for the wave function for stationary states
$\psi\left(\overrightarrow(r)\right)$- the function must be at all points:
- continuous,
- unambiguous,
- finite.
If potential energy has a discontinuous surface, then on such surfaces the function $\psi\left(\overrightarrow(r)\right)$ and its first derivative must remain continuous. In the region of space where potential energy becomes infinite, $\psi\left(\overrightarrow(r)\right)$ must be zero. The continuity of the function $\psi\left(\overrightarrow(r)\right)$ requires that at any boundary of this region $\psi\left(\overrightarrow(r)\right)=0$. The continuity condition is imposed on the partial derivatives of the wave function ($\frac(\partial \psi)(\partial x),\ \frac(\partial \psi)(\partial y),\frac(\partial \psi)(\ partial z)$).
Example 1
Exercise: For a certain particle, a wave function of the form is given: $\psi=\frac(A)(r)e^(-(r)/(a))$, where $r$ is the distance from the particle to the center of force (Fig. 1 ), $a=const$. Apply the normalization condition, find the normalization coefficient A.
Picture 1.
Solution:
Let us write the normalization condition for our case in the form:
\[\int((\left|\psi\right|)^2dV=\int(\psi\psi^*dV=1\left(1.1\right),))\]
where $dV=4\pi r^2dr$ (see Fig. 1 From the conditions it is clear that the problem has spherical symmetry). From the conditions of the problem we have:
\[\psi=\frac(A)(r)e^(-(r)/(a))\to \psi^*=\frac(A)(r)e^(-(r)/(a ))\left(1.2\right).\]
Let us substitute $dV$ and wave functions (1.2) into the normalization condition:
\[\int\limits^(\infty )_0(\frac(A^2)(r^2)e^(-(2r)/(a))4\pi r^2dr=1\left(1.3\ right).)\]
Let's carry out the integration on the left side:
\[\int\limits^(\infty )_0(\frac(A^2)(r^2)e^(-(2r)/(a))4\pi r^2dr=2\pi A^2a =1\left(1.4\right).)\]
From formula (1.4) we express the required coefficient:
Answer:$A=\sqrt(\frac(1)(2\pi a)).$
Example 2
Exercise: What is the most probable distance ($r_B$) of an electron from the nucleus if the wave function that describes the ground state of the electron in a hydrogen atom can be defined as: $\psi=Ae^(-(r)/(a))$, where $ r$ is the distance from the electron to the nucleus, $a$ is the first Bohr radius?
Solution:
We use a formula that determines the probability of the presence of a microparticle in a volume $dV$ at time $t$:
where $dV=4\pi r^2dr.\ $Hence, we have:
In this case, we write $p=\frac(dP)(dr)$ as:
To determine the most probable distance, the derivative $\frac(dp)(dr)$ is equal to zero:
\[(\left.\frac(dp)(dr)\right|)_(r=r_B)=8\pi rA^2e^(-(2r)/(a))+4\pi r^2A^ 2e^(-(2r)/(a))\left(-\frac(2)(a)\right)=8\pi rA^2e^(-(2r)/(a))\left(1- \frac(r)(a)\right)=0(2.4)\]
Since the solution $8\pi rA^2e^(-(2r_B)/(a))=0\ \ (\rm at)\ \ r_B\to \infty $ does not suit us, it goes like this:
> Wave function
Read about wave function and probability theories of quantum mechanics: the essence of the Schrödinger equation, the state of a quantum particle, a harmonic oscillator, a diagram.
We are talking about the probability amplitude in quantum mechanics, which describes the quantum state of a particle and its behavior.
Learning Objective
- Combine the wave function and the probability density of identifying a particle.
Main points
- |ψ| 2 (x) corresponds to the probability density of identifying a particle in a specific place and moment.
- The laws of quantum mechanics characterize the evolution of the wave function. The Schrödinger equation explains its name.
- The wave function must satisfy many mathematical constraints for computation and physical interpretation.
Terms
- The Schrödinger equation is a partial differential characterizing a change in the state of a physical system. It was formulated in 1925 by Erwin Schrödinger.
- A harmonic oscillator is a system that, when displaced from its original position, is influenced by a force F proportional to the displacement x.
Within quantum mechanics, the wave function reflects the probability amplitude that characterizes the quantum state of a particle and its behavior. Usually the value is complex number. The most common symbols for the wave function are ψ (x) or Ψ(x). Although ψ is a complex number, |ψ| 2 – real and corresponds to the probability density of finding a particle in a specific place and time.
Here the trajectories of the harmonic oscillator are displayed in classical (A-B) and quantum (C-H) mechanics. The quantum ball has a wave function displayed with the real part in blue and the imaginary part in red. TrajectoriesC-F – examples of standing waves. Each such frequency will be proportional to the possible energy level of the oscillator
The laws of quantum mechanics evolve over time. The wave function resembles others, such as waves in water or a string. The fact is that the Schrödinger formula is a type of wave equation in mathematics. This leads to the duality of wave particles.
The wave function must comply with the following restrictions:
- always final.
- always continuous and continuously differentiable.
- satisfies the appropriate normalization condition for the particle to exist with 100% certainty.
If the requirements are not satisfied, then the wave function cannot be interpreted as a probability amplitude. If we ignore these positions and use the wave function to determine observations of a quantum system, we will not get finite and definite values.