In quantum mechanics, the SchrÃÆ'¶dinger equation is a mathematical equation that describes the change over time of the physical system in which quantum effects, such as wave-particle duality, are significant. These systems are called quantum (mechanical) systems. This equation is considered a central outcome in the study of quantum systems, and this derivation is an important milestone in the development of the theory of quantum mechanics. It was named after Erwin SchrÃÆ'¶dinger, who derived the equation in 1925, and published in 1926, forming the basis for his work which resulted in him being awarded the Nobel Prize in Physics in 1933.
In classical mechanics, Newton's second law ( F = m a ) is used to create a mathematical prediction for what path given by the physical system will take time after a set of known initial conditions. Solving this equation gives the position, and momentum of the physical system as a function of the external force F on the system. Both parameters are sufficient to describe the situation at any instant moment. In quantum mechanics, Newton's analogous analogs are Schrödinger's equations for quantum systems (usually atoms, molecules, and subatomic particles are free, bound, or localized). The equation is mathematically described as a linear partial differential equation, which describes the time-evolution of the system's wave function (also called "state function").
The concept of wave function is a fundamental postulate of quantum mechanics, which defines the system status at every spatial position, and time. By using these postulates, the SchrÃÆ'¶dinger equation can be derived from the fact that the time evolution operator must unite, and therefore must be generated by the exponential self-adjoint operator, which is a quantum Hamiltonian. The derivation is described below.
In Copenhagen's interpretation of quantum mechanics, the wave function is the most complete description that can be given from the physical system. Solutions for the Schrödinger equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, perhaps even the entire universe. The Schrödinger equation is central to all applications of quantum mechanics including quantum field theory combining special relativity with quantum mechanics. Quantum gravity theory, like string theory, also does not change the Schrödinger equation.
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions, since there are other quantum mechanical formulations such as matrix mechanics, introduced by Werner Heisenberg, and the formulation of road integrals, developed mainly by Richard Feynman. Paul Dirac combines matrix mechanics and Schrödinger equations into a single formulation.
Video Schrödinger equation
Equation
The time-dependent equation
The form of the Schrödinger equation depends on the physical situation (see below for specific cases). The most common form is the time-dependent Schrö¶dinger equation (TDSE), which gives an overview of the system evolving over time:
di mana i adalah unit imajiner, ? adalah konstanta Planck yang berkurang yaitu: , simbol ? / ? t menunjukkan turunan parsial terhadap waktu t , ? (huruf Yunani psi) adalah fungsi gelombang dari sistem kuantum, r dan t adalah vektor posisi dan waktu masing-masing, dan ? adalah operator Hamiltonian (yang mencirikan total energi sistem yang sedang dipertimbangkan).
The best known example is the nonrelativistic Schrö¶dinger equation for the relative co-ordinates of a single particle moving in the electric field of a second particle, usually much heavier, but not a magnetic field, see Pauli equation):
where ? is a "diminished mass particle", V is potential energy, ? 2 is Laplacian (differential operator), and ? is a wave function (more precisely, in this context, it is called a "position-space wave function"). In plain language, it means "total energy is equal to kinetic energy plus potential energy", but the term takes an unknown form for the reasons described below.
Given the particular differential operators involved, this is a linear partial differential equation. It is also a diffusion equation, but unlike the heat equation, this one is also the wave equation given by imaginary units that exist in transient terms.
The term "SchrÃÆ'¶dinger equation" may refer to a general equation, or a specific nonrelativistic version. This general equation is quite common, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by incorporating various expressions for the Hamiltonian. The specific nonrelativistic version is a strictly classical approach to reality and produces accurate results in many situations, but only to a limited extent (see relativistic quantum mechanics and relativistic quantum field theory).
To apply the Schrödinger equation, the Hamiltonian operator is set for the system, accounting for the kinetic energy and potential of the particles that make up the system, then incorporated into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.
The time-independent equation
The time-dependent Schrö¶dinger equation described above predicts that the wave function can form a standing wave, called stationary status (also called "orbital", as in atomic orbitals or molecular orbitals). These countries are so important as their individual studies then simplify the task of solving the Schrö¶dinger equations depending on the time for each country. The stationary status can also be described with a simpler form of the Schrödinger equation, the time-free Schrö¶dinger equation (TISE).
where E is a constant equal to the total energy of the system. This is only used when the Hamiltonians themselves do not rely on time explicitly. However, even in this case the total wave function still has a time dependency.
In words, the equation states:
-
- When a Hamiltonian operator acts on a specific wave function ? , and the result is proportional to the same wave function ? , then ? is a stationary state, and the proportionality constant, E , is the energy of the state? .
In the linear algebra terminology, this equation is an eigenvalue equation and in this sense the wave function is the eigen function of the Hamiltonian operator.
As before, the most common manifestation is the nonrelativistic Schrö¶dinger equation for one particle moving in an electric field (but not a magnetic field):
with the definition as above.
The time-independent Schrö¶dinger equation is discussed further below.
Maps Schrödinger equation
Derivation
So far, H is just an abstract Hermitian operator. However using the principle of correspondence is possible to show that, within the classical limits, the expected value of H is indeed the classical energy. The principle of correspondence does not completely improve the form of quantum Hamiltonian because of the principle of uncertainty and therefore the exact form of quantum Hamiltonian must be improved empirically.
Implications
The SchrÃÆ'¶dinger equation and its solution introduce a breakthrough in thinking about physics. The Schrödinger equation is the first of its kind, and the solution causes very unusual and unpredictable consequences for the moment.
Total, kinetic, and potential energy
The overall form of the equation is not unusual or unexpected, because it uses the principle of conservation of energy. The term non-relativistic Schrö¶dinger equation can be defined as the total energy of the system, equal to the kinetic energy of the system plus the potential energy of the system. In this case, just as in classical physics.
Quantization
The SchrÃÆ'¶dinger equation predicts that if certain properties of a system are measured, the results may be quantized , which means that only certain discrete values ​​can occur. One example is the quantization of energy : the energy of electrons in an atom is always one of the quantized energy levels, a fact that is found through atomic spectroscopy. (Quantization of energy is discussed below.) Another example is the quantization of angular momentum. This is the assumption in the previous Bohr atom model, but this is prediction of the Schrödinger equation.
Another result of the SchrÃÆ'¶dinger equation is that not every measurement provides quantized results in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value in a continuous range.
Measurement and uncertainty
In classical mechanics, a particle has, at all times, the right position and the right momentum. These values ​​change deterministically when the particle moves according to Newton's law. Under Copenhagen's interpretation of quantum mechanics, particles do not have precisely defined properties, and when they are measured, the results are drawn at random from the probability distribution. The SchrÃÆ'¶dinger equation predicts what the probability distribution is, but basically can not predict the exact result of any measurement.
The Heisenberg uncertainty principle is a statement of measurement uncertainty inherent in quantum mechanics. It states that the more precisely the position of the particle is known, the less the momentum is known, and vice versa.
The SchrÃÆ'¶dinger equation describes the (deterministic) evolution of the wave function of a particle. However, even if the wave function is known for certain, the specific measurement results on the wave function are uncertain.
Quantum tunneling
In classical physics, when a ball is rolled slowly up a large hill, the ball stops and spins back, for it does not have enough energy to pass through the top of the hill to the other side. However, the Schrödinger equation predicts that there is little chance that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called a quantum tunnel. This is related to the distribution of energy: although the position of the assumption of the ball seems to be on one side of the hill, there is the possibility of finding it on the other side.
Particles as waves
The nonrelativistic Schrö¶dinger equation is a type of partial differential equation called the wave equation. Therefore, it is often said that particles can exhibit behaviors normally associated with waves. In some modern interpretations, this description is reversed - quantum states, ie waves, are the only genuine physical reality, and under appropriate conditions, it can exhibit particle-like behavioral traits. However, Ballentine points out that such an interpretation has a problem. Ballentine suggests that while it can be debated to associate physical waves with a single particle, there are still only one Schrödinger wave equation for many particles. He points out:
- "If the physical wave field is associated with the particle, or if the particle is identified with the wave packet, then corresponding to the N particle interacting there must be N waves interacting in the ordinary three-dimensional space But according to (4.6) is not the case, but there is a "wave" function in the abstract 3N-dimensional configuration space.The misinterpretation of psi as a physical wave in ordinary space is possible only because the most common application of quantum mechanics is the state of a single particle, where the configuration space and ordinary space are isomorphic. "
Two-slit diffraction is a well-known example of strange behavior that is displayed on a regular basis, which is not intuitively related to particles. The overlapping waves of the two slits cancel each other in several locations, and reinforce each other in other locations, causing complicated patterns to emerge. Intuitively, one would not expect this pattern from firing a particle in a gap, because particles must pass through one gap or another, not overlapping the second complex.
However, since the SchrÃÆ'¶dinger equation is a wave equation, a single particle fired through a double slit does not show this same pattern (image at right). Note: Experiments must be repeated many times for complicated patterns to appear. Although this is counter-positive, the prediction is correct; in particular, electron diffraction and neutron diffraction are well understood and widely used in science and engineering.
Associated with diffraction, the particle also displays superposition and interference.
The superposition properties allow the particles to be in a quantum superposition of two or more quantum states at the same time. However, it should be noted that the "quantum state" in quantum mechanics means probability that the system will, for example in the x position, does not mean the system will be in position x . That does not imply that the particle itself may be in two classical states at once. Indeed, quantum mechanics generally can not assign values ​​to property before measurement at all.
Multi-World Interpretation
In Dublin in 1952, Erwin SchrÃÆ'¶dinger gave a lecture in which at one point he hastily warned his listeners that what he was about to say might be "looking crazy". It is that, when the equations seem to illustrate several different histories, they are "not alternatives but they all actually happen together". This is the earliest known reference to the interpretation of Many-world quantum mechanics.
Interpretation of wave function
The SchrÃÆ'¶dinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what , exactly, the wave function. The interpretation of quantum mechanics answers the question as to what is the relationship between the wave function, the underlying reality, and the results of the experimental measurements.
An important aspect is the relationship between the Schrödinger equation and the collapse of the wave function. In the oldest Copenhagen interpretation, the particles follow the Schrödinger equation except during the collapse of the wave function, in which they behave completely differently. The emergence of quantum decoherence theory allows alternative approaches (such as the interpretation of many worlds and a consistent history), where the Schrödinger equation is always always fulfilled, and the collapse of the wave function must be explained as a consequence of the Schrödinger equation.
Historical background and development
After the quantization of Max Planck's light (see black body radiation), Albert Einstein interprets the Planck quanta into photons, particles of light, and proposes that the energy of the photon is proportional to its frequency, one of the first signs of wave-particle duality.. Since energy and momentum are related in the same way as frequencies and wave numbers in special relativity, it follows that the p < ? , or is proportional to the wave number k :
where h are Planck constants and ? is the reduced Planck constant, h/2? . Louis de Broglie hypothesizes that this is true for all particles, even particles that have masses such as electrons. He points out that, assuming that the material waves are spreading along with their particles, the electrons form a standing wave, which means that only certain discrete rotation frequencies about the nucleus are allowed. This quantized orbit corresponds to different energy levels, and de Broglie reproduces the Bohr model formula for energy levels. The Bohr model is based on quantization assumed angular momentum L according to:
Menurut de Broglie, elektron dideskripsikan oleh gelombang dan sejumlah panjang gelombang harus sesuai sepanjang keliling orbit elektron:
This approach essentially limits the wave of electrons in one dimension, along the orbit of the radius of the radius r .
In 1921, before de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of 4-vector momentum energy relativity to get what we now call the de Broglie relation. Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve its energy eigenvalues ​​for the hydrogen atom. Unfortunately the paper was rejected by Physical Review, as told by Kamen.
Following up on de Broglie's idea, physicist Peter Debye commented that if particles behave like waves, they must satisfy some wave equations. Inspired by Debye's remarks, Schrödinger decided to find the correct 3-dimensional wave equation for electrons. He is guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the optical zero-length wavelength resembles a mechanical system - the path of light beam becomes a sharp imprint that adheres to the Fermat principle, analogous to the principle of minimum action. The modern version of the reason is reproduced below. The equations he found were:
However, at that time, Arnold Sommerfeld has perfected Bohr's model with relativistic correction. Schrà bledinger relates the relativity of energy momentum to find what is now known as the Klein-Gordon equation in the potential of Coulomb (in natural units):
He found a standing wave of this relativistic equation, but the relativistic correction does not correspond to the Sommerfeld formula. Desperately, he removed his calculations and alienated himself in an isolated mountain hut in December 1925.
While in the cabin, Schrüdinger decided that the previous nonrelativistic calculations were new enough to be published, and decided to leave the problem of relativistic correction for the future. Despite difficulties in solving differential equations for hydrogen (he sought help from his friend mathematician Hermann Weyl) SchrÃÆ'¶dinger showed that his nonrelativistic version of the wave equation produced the correct hydrogen spectral energy in a paper published in 1926. equation, SchrÃÆ'¶dinger count series spectral hydrogen by treating hydrogen atomic electrons as waves ? ( x , t ) , moving in the potential of either V by proton. This calculation accurately reproduces the energy level of the Bohr model. In a paper, SchrÃÆ'¶dinger himself describes this equation as follows:
The 1926 paper was enthusiastically supported by Einstein, who saw the wave-matter as an intuitive portrayal of nature, as opposed to Heisenberg's matrix mechanics, which he considered too formal.
The SchrÃÆ'¶dinger equation specifies the behavior ? but did not say anything about its nature . SchrÃÆ'¶dinger tried to interpret it as a load density in his fourth paper, but he did not succeed. In 1926, just days after the fourth and final Schrödinger paper was published, Max Born succeeded in interpreting ? as the probability amplitude, whose square is absolutely equal to the probability density. SchrÃÆ'¶dinger, though, has always opposed a statistical or probabilistic approach, with related discontinuities - such as Einstein, who believed that quantum mechanics was a statistical approach to the underlying deterministic theory - and never reconciled with Copenhagen's interpretation.
Louis de Broglie in his later years proposed a real-valued wave function that connected to complex wave functions with proportionality constant and developed the De Broglie-Bohm theory.
The wave equation for particles
SchrÃÆ'¶dinger equation is a diffusion equation, solution is a function that describes wave-like motion. The wave equation in physics can usually be derived from other laws of physics - the wave equation for mechanical vibrations in strings and matter can be derived from Newton's law, where the wave function represents the transfer of matter, and the electromagnetic waves of the Maxwell equations, where the wave function is the electric field and magnet. The basis for the SchrÃÆ'¶dinger equation, on the other hand, is the energy of systems and propositions separate from quantum mechanics: the wave function is the description of the system. The SchrÃÆ'¶dinger equation is therefore a new concept in itself; as Feynman says:
The foundation of this equation is structured into linear differential equations based on classical energy conservation, and is consistent with the De Broglie relationship. The solution is the wave function ? , which contains all known information about the system. In the Copenhagen interpretation, modulus ? is related to the probability of particles in some spatial configurations at some instant time. Solve equations for ? can be used to predict how the particles will behave under a specified potential influence and with each other.
The SchrÃÆ'¶dinger equation is developed primarily from the De Broglie hypothesis, a wave equation that will describe particles, and can be constructed as shown informally in the following sections. For a more rigorous description of the Schrödinger equation, see also Resnick et al .
Consistency with energy conservation
Energi total E dari sebuah partikel adalah jumlah energi kinetik T dan energi potensial V , jumlah ini juga merupakan ekspresi yang sering untuk Hamiltonian H dalam mekanika klasik:
Secara eksplisit, untuk partikel dalam satu dimensi dengan posisi x , massa m dan momentum p , dan energi potensial V yang umumnya bervariasi dengan posisi dan waktu t :
Untuk tiga dimensi, vektor posisi r dan vektor momentum p harus digunakan:
This formalism can be extended to a certain number of particles: the total energy of the system is the total kinetic energy of the particles, plus the total potential energy, again the Hamiltonian. However, there can be an interaction between the particles (problem N ), so the potential energy V may change as the spatial configuration of the particles change, and possibly over time. The potential energy, in general, is not the sum of the separate potential energy for each particle, it is the function of all the spatial positions of the particles. Explicitly:
Linearity
Fungsi gelombang yang paling sederhana adalah gelombang bidang bentuk:
where A is the amplitude, k wavevector, and ? angular frequency, of the plane wave. In general, impure physical situations are portrayed by field waves, so for generality the principle of superposition is required; each wave can be made with the superposition of a sinusoidal plane wave. So if the equation is linear, the linear combination of the field wave is also a permissible solution. Then the necessary and separate requirement is that the Schrödinger equation is a linear differential equation.
Untuk diskrit k jumlahnya adalah superposisi dari gelombang bidang:
untuk beberapa koefisien amplitudo nyata A n , dan untuk terus k , jumlah menjadi integral, Transformasi Fourier dari fungsi gelombang ruang momentum:
where d 3 k = dk x dk y dk z is a variable volume element in k -space, and the integral is taken over all < b> k -space. Wave momentum function ? ( k ) appears in the integrand because the position and momentum of the space wave function are fourier transforms to each other.
Consistency with De Broglie relationship
Hipotesis quanta cahaya Einstein (1905) menyatakan bahwa energi E dari foton sebanding dengan frekuensi ? ( atau frekuensi sudut, ? = 2 ? ? ) dari wavepacket cahaya kuantum yang sesuai:
Likewise, the De Broglie hypothesis (1924) states that each particle may be associated with a wave, and that the momentum
of the particles is inversely proportional to the wavelength ? of such waves (or comparable to the number of waves, k = 2? / ? ), in one dimension, by:
sementara dalam tiga dimensi, panjang gelombang ? terkait dengan besarnya wavevector k :
The Planck-Einstein and de Bro
Source of the article : Wikipedia
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