The operator of force

1. The observability of quantum system is described by a linear Hermitian operator, which is a basic assumption of quantum mechanics. The mechanical quantity operator has all the properties of Hermite operator. For example, the average value of Hermite operator must be real. You can refer to Chapter 3 of the second edition of Science Press in the course of quantum mechanics

2. The states in quantum mechanics satisfy the superposition principle, which naturally endows them with the mathematical structure of linear space. According to nott's theorem, every continuous symmetric transformation of the system (that is, the transformation that does not change the physical structure of the system and does not affect the experimental / measurement results) corresponds to a conserved quantity Q. under these symmetric transformations, the change of the system state is of course described by a matrix (or operator), which has the form of e ^ (- ith), Where t is a matrix corresponding to this kind of transformation, which is called the generator of this kind of transformation, and H is a continuous parameter of this kind of transformation. Suppose that the value of a certain physical quantity Q can be taken as Q1, Q2, Q3... Generally speaking, the value of Q is uncertain after the system is measured, but when the system is in some states, the result of measuring q is certain| q2>,| q3>,...... To mark these states. Let the symmetric transformation corresponding to Q be e ^ (- ith), then when the system is in - say - | Q1 & gt; If q is measured again after transformation, the result is still Q1, that is to say, the system is still in | Q1 & gt; Therefore, e to the continuity of parameter h, | Q1 & gt; Is the eigenvector of the operator t. T is in the form of | Q1 & gt| q2>,| q3>,...... Obviously, the diagonal elements can only be related to Q1, Q2, Q3... That is to say, the physical quantity Q is expressed by the operator T, and the eigenvalue of T represents the value of Q.

3. In quantum mechanics, when a micro particle is in a certain state, its mechanical quantities (such as coordinates, momentum, angular momentum, energy, etc.) generally do not have definite values, but have a series of possible values, each of which appears with a certain probability. When the state of the particle is determined, the probability that the mechanical quantity has a certain possible value is completely determined. For example, when an electron in a hydrogen atom is in a certain bound state, its coordinate and momentum have no definite value, but the probability that the coordinate has a certain value R or momentum has a certain value is completely definite. These characteristics of mechanical quantities in quantum mechanics are not found in classical mechanics. In order to reflect these characteristics, operators are introced to express mechanical quantities in quantum mechanics

an operator is a symbol that performs some mathematical operation on a wave function. Add & quot to the words representing mechanical quantities; ∧" Sign to represent the operator of this mechanical quantity. Such as coordinate operator, momentum operator. When the state of a particle is described by the wave function (R,), the effect of the coordinate operator on the wave function is r multiplied by (R,), and the effect of the momentum operator on the wave function is differential

4. 1. In quantum mechanics, the mechanical quantity is expressed by an operator, denoted as fhat (i.e. f has a tip on its head, pronounced hat, abbreviated as f)
2. * (star) denotes complex number or conjugate of state vector. Generally, it is also represented by complex number with a bar, that is, the real part of complex number is invariant and the imaginary part is inverse. If it is represented by Dirac sign, then state a can be written as right vector | A & gt;, Its complex conjugate a * can be written as left vector & lt; a|< br />3. † The Hermitian conjugation of an operator is called dagger, which is defined as (U, F & 8224; v) = (Fu, V), "()" denotes inner proct
4. If the Hermitian conjugate of an operator is equal to itself, i.e. F & 8224; = F then this operator is called Hermitian operator. The operators representing mechanical quantities are all Hermitian operators. For bounded operators, Hermitian property and self adjoint property are equivalent, while for some unbounded operators, self adjoint property is stronger than Hermitian property. The reason is that the self adjoint operator also requires its basis vector to form a complete system There are many discussions on the difference between Hermitian and self adjoint on the Internet. Generally, they are treated equally.)
5. Operators can also be represented by matrices. Every element of a matrix is a complex number. For a matrix, its Hermitian conjugate is equivalent to transposing every element into a complex conjugate. In quantum mechanics, it is meaningless to carry out only complex conjugate or transpose transformation on a matrix. Hermite operator corresponds to Hermite matrix, that is, conjugate transpose equals itself< Hermite matrix is a generalization of symmetric matrix in complex field, because symmetric matrix can do orthogonal transformation with orthogonal matrix; Similarly, Hermitian matrix can also use unitary matrix to carry out unitary transformation, that is, the transformation of mechanical quantities between different images. The definition of unitary operator is an inner proct preserving operator whose corresponding unitary matrix satisfies Hermitian conjugate and its inverse, that is, UU & 8224; = I
7. Hermite operator is actually a kind of mapping of Hilbert space (complex vector space), which is the generalization of second-order tensor (mapping of real vector space) in complex vector space. In essence, they are all a kind of mapping, or transformation
8. All invertible operators (or corresponding matrices) form a general (complex) linear group, and all unitary operators form a unitary group; They are the generalization of general (real) linear group and orthogonal group in complex vector space.

5. The problem is a bit confusing. Do you mean the sign of partial derivative and wave function? Besides, there are not so many cases you mentioned

6.

This is one of the five basic hypotheses of quantum mechanics. Corresponding to item 3 below. Let me explain to you

First of all, quantum mechanics is described in Hilbert space. The eigenvalues of Hermitian operators are real numbers, not imaginary numbers. Any observable measure must be real. You can't observe imaginary numbers, can you? Therefore, the observable operator must be Hermitian operator, and the transposed complex conjugate is equal to itself

The theoretical framework of quantum mechanics consists of the following five hypotheses:

1. the motion state of microscopic system is described by the corresponding normalized wave function There is a definite commutation relation between them, which is called quantum condition; The corresponding relationship between the three rectangular coordinate system components of the coordinate operator and the three rectangular coordinate system components of the momentum operator is called the basic quantum condition; The mechanical operator is determined by its corresponding quantum conditions. The wave function of the identical multiparticle system is symmetric for any pair of particle exchanges: the wave function of boson system is symmetric, and the wave function of fermion system is antisymmetric

7. This question can be exchanged, not answered. Let me talk about my understanding. The comparison of mechanical quantities shows that two physical quantities can be accurately measured at the same time, that is, two physical quantities can take eigenvalues in the same representation at the same time; On the contrary, non commutation indicates that two physical quantities cannot be measured accurately at the same time, that is, they cannot take eigenvalues in the same representation. From a deeper level, the commutation of two mechanical quantities shows that these two physical quantities can form a complete set of mechanical quantities. Generally, three pairs of mechanical quantities are taken to form a complete set of mechanical quantities. The eigenstates of this complete set can represent all the quantum states of Hilbert space, that is, the mechanical quantities of commutation constitute the whole world. If there is a problem that can be discussed, because the meaning of quantum mechanics itself is not as deep as people know classical mechanics, and the ultimate aspect of the problem can not be fully expressed. This is the understanding of different opinions, but its essence should remain unchanged.

8. Quantum mechanics is different from classical physics. In classical mechanics, the mechanical quantity is expressed by function, while in quantum mechanics, the motion state of micro particles is expressed by wave function, and the observable mechanical quantity needs to be obtained by applying the mechanical quantity operator on the wave function (the mechanical quantity operator can be regarded as an instrument to measure the state function), and the result of the action is the value of the mechanical quantity (eigenvalue or average value).

9. The corresponding wave function should be the intrinsic wave function if the operator of the mechanical quantity remains unchanged.