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**Biju Patnaik University of Technology BPUT - BPUT**- 4 Topics
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Postulate 3. In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues , which satisfy the eigenvalue equation This postulate captures the central point of quantum mechanics--the values of dynamical variables can be quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states). If the system is in an eigenstate of with eigenvalue , then any measurement of the quantity yield . Postulate 4. If a system is in a state described by a normalized wave function observable corresponding to will , then the average value of the is given by Postulate 5. The wavefunction or state function of a system evolves in time according to the time-dependent Schrödinger equation The central equation of quantum mechanics must be accepted as a postulate Postulate 6. The total wave function must be anti symmetric with respect to the interchange of all coordinates of one fermions’ with those of another. Electronic spin must be included in this set of coordinates. The Pauli Exclusion Principle is a direct result of this anti symmetry principle. de-Broglie's concept and uncertainty principle de-Broglie’s concept In 1924, Lewis de-Broglie proposed that matter has dual characteristic just like radiation. His concept about the dual nature of matter was based on the following observations:(a) The whole universe is composed of matter and electromagnetic radiations. Since both are forms of energy so can be transformed into each other. (b) The matter loves symmetry. As the radiation has dual nature, matter should also possess dual character. According to the de Broglie concept of matter waves, the matter has dual nature. It means when the matter is moving it shows the wave properties (like interference, diffraction etc.) are associated with it and when it is in the state of rest then it shows particle properties. Thus the matter has dual nature. The waves associated with moving particles are matter waves or de-Broglie waves. Wavelength of de-Broglie waves: Consider a photon whose energy is given by E= hυ= hc/λ – – (1)

If a photon possesses mass (rest mass is zero), then according to the theory of relatively , its energy is given by E=mc2 – – (2) From (1) and (2),we have λ = h/p = h/mc ------------- (3) If instead of a photon, we consider a material particle of mass m moving with velocity v, then the momentum of the particle, p=mv. Therefore, the wavelength of the wave associated with this moving particle is given by: h/mv λ = h/p (But here p = mv) --------(4) This wavelength is called DE-Broglie wavelength. Special Cases: 1. de-Broglie wavelength for material particle: If E is the kinetic energy of the material particle of mass m moving with velocity v, then E=1/2 mv2=1/2 m2v2= p2/2m Or p=√2mE Therefore by putting above equation in equation (4), we get de-Broglie wavelength equation for material particle as: λ = h/√2mE – – (5) 2. de-Broglie wavelength for an accelerated electron: Suppose an electron accelerates through a potential difference of V volt. The work done by electric field on the electron appears as the gain in its kinetic energy That is E = eV Also E = p2/2m Where e is the charge on the electron, m is the mass of electron and v is the velocity of electron, then Comparing above two equations, we get: eV= p2/2m Or p = √2meV Thus by putting this equation in equation (4), we get the the de-Broglie wavelength of the electron as λ = h/√2meV= 6.63 x 10-34/√2 x 9.1 x 10-31 x1.6 x 10-19 V λ=12.25/√V Å This is the de-Broglie wavelength for electron moving in a potential difference of V volt. Uncertainty Principle: In classical physics it is generally assumed that position and momentum of a moving object can be simultaneously measured exactly i.e. no uncertainties are involved in its description. But in microscopic world it is not possible. It is found that however refined our instruments there is a fundamental limitation to the accuracy with which the position and velocity of microscopic particle can be known simultaneously. This limitation was expressed by a German physicist Werner Heisenberg in 1927 and known as 'Heisenberg's uncertainty principle'. In microscopic particles we can observe two type of uncertainties viz. Uncertainty in position and momentum Uncertainty in energy and time STATEMENT: It is impossible to determine both position and momentum of an electron simultaneously. If one quantity is known then the determination of the other quantity will become impossible MATHEMATICAL REPRESENTATION: Let ∆x = uncertainty in position ∆P = uncertainty in momentum According to Heisenberg's uncertainty principle: The product of the uncertainty in position and the uncertainty in momentum is in the order of an amount involving h, which is Planck’s constant

P x x h/4 i v x x ≥ h/4пm ----------- (ii) RESULTS OF UNCERTAINTY PRINCIPLE: It is impossible to chase an electron around the nucleus. The principle describes the incompleteness of Bohr's atomic theory. According to Heisenberg's uncertainty principle there is no circular orbit around the nucleus. Exact position of an electron can not be determined precisely. SIGNIFICANCE: Heisenberg's uncertainty principle is not applicable in our daily life. It is only applicable on micro objects i.e. subatomic particles. The reason why the uncertainty principle is of no importance in our daily life is that Planck's constant 'h' is so small (6.625 x 10-34joule-seconds) that the uncertainties in position and momentum of even quiet small (not microscopic objects) objects are far too small to be experimentally observed. For microscopic phenomena such as atomic processes, the displacements and momentum are such that the uncertainty relation is critically applicable. Introduction to Schrodinger Wave Equation: Schrödinger Wave Equation: 1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of electron. (2) In it electron is described as a three dimensional wave in the electric field of a positively charged nucleus. (3) The probability of finding an electron at any point around the nucleus can be determined by the help of Schrodinger wave equation which is, ∂2Ψ/∂x2 + ∂2Ψ/∂y2 + ∂2Ψ/∂Z2 + 8π2m (E-V)Ψ /h2 = 0 Where x,y, and z are the 3 space co-ordinates, m = mass of electron, h = Planck’s constant, E = Total energy, V = potential energy of electron, Ψ = amplitude of wave also called as wave function, ∂ = for an infinitesimal change. The Schrodinger wave equation can also be written as, ∇2Ψ + (8π2m/h2) (E-V) Ψ = 0 Where ∇ = laplacian operator. Physical significance of Ψ and Ψ2 (i) The wave function Ψ represents the amplitude of the electron wave. The amplitude Ψ is thus a function of space co-ordinates and time i.e. Ψ = Ψ (x, y, z.....t) (ii) For a single particle, the square of the wave function (Ψ2) at any point is proportional to the probability of finding the particle at that point. (iii) If Ψ2 is maximum than probability of finding e- is maximum around nucleus and the place where probability of finding e- is maximum is called electron density, electron cloud or an atomic orbital. It is different from the Bohr’s orbit. (iv) The solution of this equation provides a set of number called quantum numbers which describe specific or definite energy state of the electron in atom and information about the shapes and orientations of the most probable distribution of electrons around the nucleus. Derivation of Schrodinger’s wave equation: Where, ψ’(x) = ∂ψ/∂x and ψ’’(x) = ∂2ψ/∂x2 Now wavelength λ and momentum p of the wave are related to each other by the following equation called de Broglie wavelength equation,

Putting this value of λ in above second order differential equation we get Total energy of electron E = Kinetic Energy [Ek] + Potential Energy [V(x)] ⇒ Ek=E-V(x), therefore, This is known as one dimensional S.W.E. S.W.E. in three dimensional form is, ∂2Ψ/∂x2 + ∂2Ψ/∂y2 + ∂2Ψ/∂Z2 + 8π2m (E-V)Ψ /h2 = 0 Particle in a box: energy levels: A general solution of s.w.e. is, Ψ(x) = A sin kx + B cos kx, For boundary conditions, we have Ψ(x) = 0 at x=0 or ψ (o) = 0 O= A sin 0 + B cos 0

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