One meaning of the wavefunction is that its magnitude measures the probability amplitude of the particle found at some point. If the wavefunction corresponds to an eigenfunction , then the probability distribution is independent of time ()and is probn(x) given by and the probability of finding the particle between and is Pn x1 x2 We can also calculate probability distributions for observable quantities in a similar manner to above. The probability distribution is characterized by two measures its – expectation value (estimate of the mean value of the distribution) and its uncertainty (spread of values around the mean). Suppose that an observable quantity has a discrete A set of possible values . Then, we can prepare identical systems, all in the a1, a2, N … same state and measure the value of in each system. In general, we will get a spread of A results. The best estimate of the quantity is its mean value <A> and the standard A deviation can be used as a measure of the spread or uncertainty in the measurements. The expectation value of an observable is the quantum-mechanical prediction for the mean A value. In a stationary state described by the wavefunction , the expectation value of an observable A is given by (9) where is the quantum-mechanical operator corresponding to The operator is A. between the two wavefunction and , and for this reason we call the right ‘sandwiched’ hand side of Equation (9) a integral. The first part of the integrand is called the sandwich bra () and the second part the ket (). Some of the dynamical properties of a system that can be calculated are shown in Table 1. Total probability Position Momentum* Kinetic Energy Potential Energy Total energy where is the Hamiltonian H
One meaning of the wavefunction is that its magnitude measures the probability amplitude of the particle found at some point. If the wavefunction corresponds to an eigenfunction , then the probability distribution probn(x) is independent of time ()and is given by and the probability Pn of finding the particle between x1 and x2 is We can also calculate probability distributions for observable quantities in a similar manner to above. The probability distribution is characterized by two measures – its expectation value (estimate of the mean value of the distribution) and its uncertainty (spread of values around the mean). Suppose that an observable quantity A has a discrete set of possible values a1, a2, … . Then, we can prepare N identical systems, all in the same state and measure the value of A in each system. In general, we will get a spread of results. The best estimate of the quantity A is its mean value <A> and the standard deviation can be used as a measure of the spread or uncertainty in the measurements. The expectation value of an observable A is the quantum-mechanical prediction for the mean value. In a stationary state described by the wavefunction , the expectation value of an observable A is given by (9) where is the quantum-mechanical operator corresponding to A. The operator is ‘sandwiched’ between the two wavefunction and , and for this reason we call the right hand side of Equation (9) a sandwich integral. The first part of the integrand is called the bra () and the second part the ket (). Some of the dynamical properties of a system that can be calculated are shown in Table 1. Total probability Position Momentum* Kinetic Energy Potential Energy Total energy where H is the Hamiltonian
schrodinger.docx