New chapter – quantum and stat approach We will start with a quick tour over QM basics in order to refresh our memory. Wave function, as we all know, contains.
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New chapter – quantum and stat approach We will start with a quick tour over QM basics in order to refresh our memory. Wave function, as we all know, contains all information about a physical system. I can be obtained by solving the Schrodinger Equation: Questions: Can the wave function be measured? does the wave function exist in the real world, or is it only an abstract mathematicla object?
In QM, observables are represented by eigenvaluesValues of observables allowed by nature are called operators The results of a single physical measurement is always an eigenvalue The state functions corresponding to eigenvalues (“pure states”) are called The set of all possible eigenvalues is calledspectrum Eigenvalues arerealnumbers eigenstates In Dirac notation, eigenstates are denoted as (in Dr. Wasserman’s text, I mean – the notation used in different textbooks may differ considerably).
Eigenvalues and eigenstates corresponding to an Ω op operator can be obtained by solving the equation: which is called theeigenvalue problem eigenvalue The subscript n represents the fact that the equation usually has many solutions – n may be finite or infinite depending on the given physical situation. The eigenfunctionsare called kets in Dirac notation. The conjugate of a “ket” is calledbra: “≤” is used as a symbol of Hermitian conjugation in Dr. Wasserman’s text
Note: operators operate on “kets” from the left, and on “bras” from the right., i.e., the operator is equal to its conjugate, then The eigenvalues of Hermitian operators are always real. Operators correspoding to observables are always Hermitian operators.
The full set of eigenfunctions constitute a basis that spans the entire space of the states that the physical system may assume. Those states are linear combinations of the eigenstates: in Dirac notation is a “scalar product” (a.k.a. “inner product”; it is the equivalent of a “dot product” of ordinary vectors). The eigenstates – as any basis in any vector space – must satisfy the orthonormality relation:
The orthonormality relation enables us to find a simple recipe For finding the expansion coefficients c n in a linear Combination representing an arbitrary state function: Note: it means that is the length of the projection of on
How to show that |a><b| is indeed an operator? It’s not a big challenge: an operator converts one function (vector) into another, right? Let’s then try, and operate on an arbitrary ket vector, call it |c> : Inner product of two vectors is a scalar, right? It is just a number. But specifically, a number of what kind? Yes, U R right!!! It’s a..................number! Call it z : COMPLEX So:
In particular, an interesting operator of such kind is this one:, made of the ket and bra of the same eigenvector. It is called the............................ PROJECTION OPERATOR Conclusion: the projection operator acting on a vector ψ returns the component of the ψ vector that is parallel to the eigenvector
Question: and if we take the sum of projection operators constructed from ALL eigenvectors comprising the basis? In other words, what we get if we take: x y j k Hint:
This is the so-called “completeness relation”, sometimes also called “closure relation”. We will need it soon! Probabilities Any state function, as we said, can be written as a linear combination of the eigenvectors comprising the basis: must be normalized, i.e., Since, by carrying outtaking the combination sum, one readily obtains:
Now, suppose that you are making a measurement of the observable associated with a Ω op operator, on a system that is in a quantum state ψ. As the result, you may obtain only one of the allowed eigenvalues. In general*, it is not possible to predict which one. But you can find the probability of obtaining a particular value. One of the GREAT POSTULATES of QM states that this probability is: Therefore, the c n coefficients are often called the “probability amplitudes” *There are some special situations in which it is possible to predict the outcome of a QM measurement – can you think of an example?
Average values (a.k.a. “expectation values”) Suppose that you perform measurements of a quantity associated with a Ω op operator, on a quantum system that at the time of each measurement is in the same state ψ. Each measurement yields an eigenvalue, but each time it may be a different one from the allowed ω n set. After collecting a sufficient number of results, you may calculate the average. Conventionally, it’s denoted as. It is possible to predict the value of this average, by using the well-known formula:
The sum from the preceding slide can be converted into an elegant compact expression. For doing that, we will need to use at one moment the eigenvalue equation: OK, let’s go ahead:
More about the expectation value: If the system is in a pure quantum state with a corresponding eigenvalue ω n, then the expectation value of Ω op is, of course, This is obvious – but it is instructive to show it by calculations: