Explore the fundamentals of quantum mechanics with a focus on the Schrödinger equation, wave functions, probability density, and normalization conditions. Dive into the concept of complex conjugates, potential energy in a box, and the derivation of wave functions in this informative guide.
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SCHRDINGER EQUATION The (time-independent) Schr dinger equation of a particle of mass m in a potential energy well V(x) is given by: The complex function is called the wave function. It determines the probability of all experimental outcomes. A wave function must be a continuous function.
COMPLEX CONJUGATE The complex conjugate of a complex number z is denoted by z*. To get the complex conjugate of a complex number, simply replaces all i by (-i). For example: z = a +ib z*= a-ib z = Aeiwt z*= Ae-iwt 2i 1+4 - 7i eiq e-iq -2i z = z*= 1+4 + 7i
WAVE FUNCTION AND PROBABILITY 2=y*y is the probability density. y It determines how likely the particle is found at x. More precisely, the probability of find the particle between a and b is given by: b 2dx P(a,b) = y a
NORMALIZATION CONDITION The probability of find the particle between a and b is given by: b 2dx P(a,b) = y a If you have one particle, then the probability of finding it somewhere between x = must be 1: + 2dx P(- ,+ ) = y =1 [normalization condition] - This condition must be satisfied for any wavefunction.
POTENTIAL ENERGY FOR A PARTICLE IN A BOX The picture on the right represents the potential energy V(x)of a box (or a square well). A particle inside the box cannot go beyond 0 and L because of the infinitely high energy. Solution to the Schr dinger equation gives:
DERIVATION OF THE WAVE FUNCTION We have a particle inside the infinite well. Outside the well (x < 0 and x > L), the wave function should vanish: y =0 for x < 0 and x > L.
CONTINUITY OF WAVE FUNCTION More precisely, we should write the solution as: 0, x < 0 or x > L Asin(knx), 0 x L yn=
MATH REMINDER sin2qdq 2 2 For example, if k =np L L k 0 where we used the fact that sin(2kL) = sin(2np) = 0. 1-cos(2q) =1 2q -1 =1 = dq 4sin(2q) 1+cos(2q) 2q +1 cos2qdq = dq 4sin(2q) : L L =1 =1 1 2kx -1 =L sin2(kx)dx sin2(kx)d(kx) 4sin(2kx) 2 k 0 0
FIND THE NORMALIZATION CONSTANT We found the wave function for the infinite square well. Now we will find the normalization constant A. yn= Asin(knx) with kn=np L + *yndx 1= yn - [normalization condition] L L 2 L 2sin2(knx)dx 2 = = A = A sin2(knx)dx A 2 0 0 2 L A =
OPERATOR CORRESPONDENCE In quantum mechanics, for every possible physical measurement (observation) you can make, there is a corresponding operator that you can use to make predictions. We will discuss how in the next slide, but here are some examples of the operators: Measurement Measurement Operator Operator x = x Position Momentum in the x-direction Energy Angular momentum in the z-direction
MEASUREMENT AND EXPECTATION VALUE According to quantum mechanics, in general you cannot predict the outcome of a particular experimental measurement. However, if you do the same measurement over many many identically prepared system, you can make a prediction about the expectation value (average) of the measurement. O = y* O ydx
MATH REMINDER b b ( h x)dx ( fg) x + = h [ ]a b= hb- ha = dh a a = f g x+ f xg f g ( ( fg) x - f x - x= (fg) - f xg) - f xg x + f g x dx = dx - ( fg) x - If either f or g becomes 0 at infinity (true for any wave functions), then we have: + + + f x [ ]- + - = dx - gdx = fg gdx - + + f g x f x dx = - gdx - -
MEASUREMENTS MUST BE REAL All physical measurements we perform in the lab must give real numbers as outcome (as opposed to complex numbers), so our expectation values must also be real. This is, however, not obvious in quantum mechanics because everything are written in complex numbers. Here is an example using momentum operator to show how we will always get real numbers in the end:
QUANTUM TUNNELING Classically, the particle does not have enough energy to climb over the wall. But what was impossible in classical mechanics is possible in quantum mechanics, with a small but finite probability. In fact, you have a very small (but finite) probability for falling through the chair you are sitting on!
