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Importance of Hilbert Space in Quantum Mechanics

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Importance of Hilbert Space in Quantum Mechanics IMPORTANCE OF HILBERT SPACE IN QUANTUM MECHANICS Ashoke Das1 and Sahin Injamamul Islam2 1Associate Professor,Department of Mathematics,Raiganj University,West Bengal, India 2Department of Mathematics,Raiganj University,West Bengal,India Abstract In this paper we have discussed about the importance of Hilbert space in Quantum Mechanics.Hilbert space has many applications in various field but in Quantum Mechanics Hilbert space is a vital component that can be denoted as the complete space of Inner product.Hilbert space plays a central role to integrate and determine the interpretation of Wave func- tion.We have been seen that for determining the material's electronic structure has used the atomic position.However,the implementation of the particular molecules delivers the accurate information of the change in vector position. A specific linear operator can be defined within all of the Hilbert space,it can be said that both are necessarily bounded. Key words Hilbert Space,Quantum Mechanics,Wave function,Atomic Position,Orthonormal 1 Introduction Hilbert Space is a significant component Quantum Mechanics and it can be de- noted as the complete space of inner product. Hilbert space can play a central role in order to determine the interpretation of the wave function.The absolute values of each of the wave function can be interpreted as the probability distri- bution functions. In the current study, a research paper is represented which can provide significant ideas about the importance of the Hilbert space within the area of quantum mechanics.We analyzes the aim, objectives as well as the purpose of the study. Moreover, the suitable research methodology, as well as results of the interpretation, is illuminated briefly.The current study aims to analyze the importance of Hilbert space in the area of Quantum mechanics. From a study, it has been found that in case, a specific linear operator can be defined within all of the Hilbert space, it can be said that both are necessarily 1 bounded.Therefore, the purpose of the current study was to obtain unbounded operators which is the exact subspace regarding Hilbert space. The main ob- jective of this paper is to identify the importance of Hilbert space in Quantum mechanics and analyze related postulates in this ground. In addition to that, this discussion focuses on evaluate external factors that affect Hilbert space question. 2 Some known results Quantum mechanics shows a physical system that is represented by a vector in a Hilbert Space. In the context of Hilbert space is composed of the inner product of an infinite-dimensional that shows a complete or closed property.In this way,it can automatically satisfy the linear based condition with completeness.As per the analysis, it demonstrates that orbit has a significant role in order to satisfy the liner-based combination and delivers quantum information. kΨk2=< Ψ j Ψ > In the context of the quantum mechanism there are several factors that have a vital role regarding the equation of Hilbert space such as j Ψ > and C j Ψ >.In which 'C' is consider as the nonzero factor.Physical state of the correspondents cannot be considered as the particular vectors for mentioning Hilbert space. If < Ψ j Ψ >=1 or kΨk2=1 then the mentioned equation can be formed in Hilbert space. On the other hand, in the context of the equation a meaningful assertion can be obtained. However, it can be stated that in the context of the statistics hypothesis has a vital role as it has a direct connection with the mentioned equation. Apart from that ei' and ' maintain a significant role in the statis- tics methodology and can be considered as real.ei' can be considered as the phase factor that can be multiplied by the normalized vectors to represents the physical state of affairs.Within ei' ,' is a real and in physical interpretation it represents normalization without changing the value.The notation of equa- tion is expected to suggest the analogy that delivers the quantum information along with the binary digit. In terms of quantum information, it defines two- dimensional properties that perfectly make a good sense in the Hilbert space. On the contrary, it can be seen that the gyroscope maintains an indicative role in the Hilbert space . The gyroscope is mainly included with spin half particles along with an angular momentum vector. This angular momentum vector has pointed any random direction in space. The atomic position is used in order to understand the electronic structure of the material.In solid-state chemistry,the atomic structure can be determined with the help of basis function and the basis sets.The localized basis function can be determined with the help of the auxiliary basis sets. The orthonormal orientation of these kinds of functions can be determined with the help of the Hilbert Space equation.The calculation of the spans of localized basis functions depends on different external parameters like the atomic position. For deter- mining the orientation of the quantum space,it is essential to incorporate the 2 non-orthogonal space equation for finding a suitable solution in this regard. According to Schr¨odinger'sequation of wave function,it can be noted that Hb n(x)= En n(x) Hilbert Space Equation where H is the Hamiltonian operator and the En represents the Eigen value. For transforming this equation in the quantum field, Hilbert space factor operator can be used. Related equation in this case can be defined in the interval of [0;L] and by considering f(0)=f(L)=0 for all n and m is R L 2 0 j n(x) j dx=1 and R L ∗ 0 n(x) m(x)dx=δnm ∗ Here n(x) is complex conjugate Quantum Equation Integration of overall function provides details about the infinite points lies in the orthogonal space. This can help to illustrate the phenomenon of the quantum field and the particle movements in this field can be demonstrated with application of the Hilbert space function. Determination of vector space with the liner basis factor helps to simplify the equation of movement of the particles in the quantum field . Consequently, the nature of the particle and their interaction with each other can be determined with this equation in a realistic way. 3 Example vector and matrices Particle in Quantum Well Vector X Wave function (x) Space vector space Hilbert Space 0 1 Linear operator matrix A= Hamiltonian H 1 0 well Eigenvalue problem Ax=λx Hwell n=En n n2π2h2 Eigen value λ1 = 1; λ2 = −1 En = 2mL ; n = 1; 2; 3::: 1 q Eigen vector X = p1 wave function (x)= 2 sin( nπx ) 1;2 2 ±1 n L L P2 ∗ R L ∗ Scalar product hX j Y i= n=1 xnYn h j φi = 0 (x)φ(x)dx Orthogonal basis hen j emi = δnm h n j mi = δnm Dimension 2 1 P2 P1 Completeness X= n=1hen j Xien = n=1h n j i n Vector components X=(he1 j Xi; he2 j Xi) = (h 1 j i; h 2 j i; :::) Table 1: Hilbert Space With Quantum theory 3 Results In the context of orthogonal projection, PV defines that it is a self-adjoint linear operator of H. PV is the orthogonal projection that is considered as the linear operator regarding self-adjoint on H. However, the nonzero state vectors j i and j 'i which are orthogonal represent and < i j 'i >=0 along with the distinct physical properties.In the present quantum, system does not provide the result simultaneously with two incompatible properties. As an example, it can be seen that a spin half particle cannot have both particles which are Sx=1/2 and Sz=1/2.. Accordingly, these molecules have the ability to allow incompatible particles along with the quantum theory. Moreover, it is seen that the adjoin point of this quantum is mainly referred to as the anti linear approaches, which is beneficial to analyze the molecules. 4 These parameters can be calculated to analyze the location of the parameters and the actual position of the quantum. However, it is found that this quantum analyze mainly defined the proper position of the quantum which is related to the physical variables. Therefore, it can be said that the quantum system and the Hermitian operation is important to identify the position of quantum variables position in the Hilbert space. According to the earlier discussion, it can be seen that the integration of this value is beneficial to calculate the quantum position in the Hilbert space. In terms of a Hermitian operation has mainly defined as the self-operator.As per the value of mentioned equation, it formulates the components of a vector of angular momentum in the Hilbert space. In this value, it highlights the frequent position of the position of vector momentum in quantum physics. As per the statistic analysis, it is seen that the Hilbert space is mainly defined as a function of the discrete variables in a systematic way. With the use of the Hermitian operators, it shows the value of quantum mechanics which is (u; v)=0 8 v2H implies to u = 0 The mentioned equation defined that there are no such Hermitian inner product that required positivity in the context of Hilbert space. kuj2=(u; u)1=2 As per the equation if H is considered as the preliminary space of Hermitian inner product in the regards of norms on H . The mentioned equation has a vital role in the estimation of the error in the finite element method. In the context of the Hilbert space the norms of H determine the specification of the momen- tum of the quantum position.
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