Quantum Mechanics Notes: Borns Interpretation of Wave Function

(Barn's interpretation of 4 (probabilistic interpretation In quantum mechanics, the state (or one of the States) of a particles is deşorbed by the wave function, 4(x,t) (75) :. It is suitable to use the amplitude funchon, 4 whose square moduluş, 1412 = 4*4, is equal to the intensity of the wave associated * The intensity of a wave vat a goven point in space is proposhonal to the probability of finding, at that point the material particle that corresponds to the wave. Born interpreted 1412 as the probabity density and 2 141 d JJS 4*4 dxdy.dg. as the probability of finding partide at a time t in the volume element 4*4 d² = 1 all space ie, Probability of finding particle over space is unity. = entre About 4, in brief 1) 4 must be finite for all values of x, y, 2. 2) 4 must be strigle valued, continuous every where and hence for each set of values x,4,3 must have one value only. 3. 4 must approach yo 4. Expect those regions where the potential energy, V(x,y) = 0, 4 must be continuous in all other regiay. 5 4= a, 4, +++ath [ pinuple of superposchon] aaan are arbatorary complex numbers. 6. Suppose two wavefunchons 4; and If have same energy states E. This is va case of degeneracy and the system is said to be degenerate state. 7. Two wave funchons, 4i(x) and 4y (2) عدم Baid to be orthogonal if the product of 4f(x) and the complex conjugate of 4; (x), ce; 4 (x) Vanishes when integrated wat x from -200 to +0. over the in torval Operators • operators are mathematical tool associated with each quantum system with each measurable Parameter is a physical system of a quantum mechanical operatog eg: For posihon operator is momentum AN -ih a 201 kinetic energy is h² 22 or ih 2m dx2 or 22 -h An operatur transform one function into anallien. (eg: I transforms the Function ginx unto coşx.. am Eigen value and Eigen funcion we know that the wave function must be single valued, continuous and should be ferite. *When am operator operates A4 = 24 on a Funchon, ve where A operator, 4 funchon, a constant Here A acty on & and the result is constant times the function Then the function is called eigon funcion and A is called eigen value. Schördinger equation duplazumaril [1-domensional wave] Phage Consider a simple harmonic wanie travelling from left to right along I've taxis with o as orgin. The parkde, p of the medicum vibrate simple harmonically It any instant,t, duplacement of the particle, y = A sincot Consider Pat x distance from O, Then desplacement of Pus, = Asin [wt-2x] = A Sinw [t-A] Using Euler's idenhty, Y=Aexp[-ico (t-✗) -iw(t-%) 4 Ae v us a function of poschon sx and time t. +V 23 4EP4+4 mulhplying both sides with 4 From eq (4) and (5), we get E4-b -b² 224 the →→ Substituting (7) $(8) (Values of E4 $ p24) on eq (6) -h 24 -h² 220 +۷۶ xe well8 tnt the 24- Ih 24 απ at = This eq. ure presents the time dependord form of Schrodingues equation In 3-dimensing, the eq, ih 24 2π 79- [ 2 24 + + +4 → 70 Fe the ih-24 = ²²²+4→(1) 2 2 m As 4 is a function of or and t ih(x,t) at = (714) ²²+ Vit) 4(.t) 2 2m ⇒ For a free Particle, V = 0 :eq(9) in 24 at 2 -h 224 2x2 2 am Also, th 2m at E= P 2m From the above, (14) cha Energy operator Eq: (11) can be rearranged as, and ax momentum operator operator ofer the Hamiltonian This is the Time dependent Schrodinger equations far a particle of mass m and potential Vcont) Due to the presence of i eq (14) well be complex Time Time - independent Schrodinger Equation. Some a Sihuahong when partide does not Potential energy depend upon time explicaty. Then the Forces aching on the particle anll be a function of position or only. The scherodinger equation so derived without any referenu to hime tus called Stationary State wave equation on hime independent ware equation. and (PCE) which depends only on time. This es caked the method of seperation of variables Here wave equat Finchon 4(rit) can be seperated into two function 400) which depend on only poschen 4(t) = 4(x) (E) (1) Time dependent Schrodinger equation s in 24 2m (2) Subshhibing eq (1) and dividing throughout by 4(0), $(1) ih da 4(4) dt 4- V () →(2) LHS as a funchon of time only, RHS is a Finchon of space coordinate Ast and or are independent variables, each sides must be equal to a constant say E Hence, do(t) $(1) 2m dt a $ E h √(x)] 4 + V(x) (6)← (1)←← (16) Solution of eg )3) = $(0) exp(-iet/n) → (5) $(0) value of f at hime, t=0 q(4) us called time independent Schrodinger equation and also called the amplitudo equation because 4(31) determines the amphhide of the werve function 4(t). on time independent Here WONE hoo function wave equation. equat hinchon 4(rit) can be seperated ento 4(0) which depend and $CE) which depends only on on only poshon hine This is called the reettiod of seperation of variables 4(9/2) = 4(1) (1) (1) Time dependent Schrodinger equation co 2m Subshhibing eq (1) and dividing throughout by 4(0), $(1) in =)²+x)] 400→(2) +V LHS is a function of time only, RHS is a finchon of spare coordinate As it and os are independent variables, each sides must be equal to Hence, 1 a constant say E (6) 42 h 2 m +(x)](9) = E4(1)(4) Solution of eq )3( وفي $(t) = $(0) exp(-iEt/n) → (5) $(0) value of of at hime, t=0 (914) us called time independent Schrodinger equation and also called the amplitudo equation because 4(3) determines 4(rit). the amphhide of the wente funchon 4(4) = 4 (21) $(0) exp(-E/n)→ E sepration constant, is the energy of the system Differentiate eg (1) witt & multiply with ib. ih 24 (0) = E4 (4) 21 Malhplying both sides of ag (7) with at Bore left and entegrate withours the limiet - to too. */4* (in) 4 (re)dt = E → LHS expectation value of E [-h/m²+Vean] is the operator equivalent to [(p²/2m) + Vcor)] which expresent the energy in the operator form. The energy operator 15 represented by H, the Hamiltonian. Thus the time ornidependored Schrodinger equation in the operator fourn looks litho, H4=E49 [ on Energy eigen value equation 4→ eigin function of the operator H E (constant) kalled the corresponding eigen value/ H→ Hamiltonion energy eign value. Most General Son of the Setanfinger equation 4(9) = A4() exp -Ent 00