Teoretisk Fysik KTH Advanced QM (SI2380), Lecture 2 (Summary of concepts) 1 PLEASE LET ME KNOW IF YOU FIND TYPOS (send email to langmann@kth.se) The laws of QM 1. I now discuss the laws of QM and their physical interpretation. To keep the mathematics simple I discuss these laws for QM systems with a finite dimensional state space, as discussed in Lecture 1. Our discussion is such that it straightforwardly generalizes to quantum systems with infinite dimensional state spaces (at least formally: there are additional mathematical subtleties then). 2. The first law is: QM Law I: States 2 of a quantum system (QS) correspond to the non-zero vectors in a Hilbert space H. I do not assume that you know what a Hilbert space is in my following discussion (but you may want to recall its definition in some mathematics book). Instead, I illustrate the essential properties of Hilbert spaces in a simple example that is adequate for QM systems with a N-dimensional state space (N a finite, positive integer). For such a system one can choose N preferred states j, j = 1, 2,..., N, and the possible states of the system are ψ = ψ 1 1 + ψ 2 2 + + ψ N N j ψ j j (1) with ψ j C for j = 1, 2,..., N with at least one ψ j non-zero. As discussed in Lecture 1, one simple interpretation of such a model would be electron moving on N atoms, and in this case j could represents the state where the electron is on site j (I put quotation marks to emphasize that this is one possible physical interpretation). 3. To describe the Hilbert space structure of the states in (1) we need some definitions. The state vector ψ in (1) is called ket, and to each such ket one assign a bra as follows, ψ ( ψ ) = ψ j j, (2) j with the bar complex conjugation, Moreover, to two state vectors ψ and φ one assigns the scalar product ψ φ = j ψ j φ j (3) 1 Version 1 (v1) by Edwin Langmann on August 22, 2010; revised by EL on September 24, 2010 2 In this paragraph state is short for pure state. We will discuss mixed states in a future lecture. 1
with the bar complex conjugation. This definition can be remembered by the following rules j k = δ jk (4) (Kronecker delta) for all j, k, ψ φ ( ψ ) φ = ψ φ, (5) and linearity. One can also remember this as follows: a bra ψ and a ket φ can be combined to a braket ψ φ. We call the vector space of all ψ in (1), including the zero vector ψ = 0 (i.e. ψ in (1) with ψ 1 = ψ 2 = = ψ N = 0), together with the scalar product in (3), H N. One can show that H N obeys the axioms of a Hilbert space. 3 4. In this case we can naturally identify element in H N and vectors in C N by the following correspondence: ψ 1 ψ 2 ψ ψ = (6). where ψ j = j ψ. Then the scalar product ψ φ is obviously equal to the usual scalar product of the corresponding vectors ψ and φ in C N : ψ φ = ψ φ with ψ = (ψ 1, ψ 2,, ψ N ). 5. It is useful to introduce the following notation: ψ 2 ψ ψ 2. Note that ψ is a norm, in particular, it is always non-negative and zero only for the zero vector ψ = 0. It is often useful to assume that a state ψ is normalized, i.e. ψ = 1. This can be done without loss of generality: for any state ψ the following is (obviously) a normalize state: 1 ψ. (7) ψ 6. The physical interpretation of the scalar product in out example is as follows: There exist special states j of the system where one can tell exactly that the electron is on atom j but on no other. However, if the system is in a general normalized state ψ, a measurement that determines if the electron is on atom j or not will give yes with probability ψ N p j = j ψ 2 (8) and no with probability 1 p j, for all j = 1, 2,..., N. Moreover, if the result of this measurement is yes. Note that j p j = 1 (prove that!): one can find the electron for sure at one of the atoms. 3 You might want to check that. 2
A good way to think about this is as follows: Suppose somebody prepares our system such that (s)he know that it is in the state φ, and then performs a position experiment to answer the question: Is the system on the atom j or not. QM says that we cannot predict this for sure: the result of this experiment can be yes or no. The best we can do is a probabilistic prediction: if you do this experiment M 1 times, then you will get p j M times the answer yes and (1 p j )M times the answer no. Only for the special states where p j is 0 or 1 one can make a certain prediction. 4 One important point if that, if in the experiment measuring if the position of the electron is on atom j the outcome is yes, then the electron IS on atom j, i.e. the system is in the state j right after such a measurement with outcome yes : the measurement affects the system! This is very different from CM where one assumes that experiments can be done so as to not affect the system. QM tells us that this is not possible even in principle. 7. What we just discussed is an example of the second QM law: QM LAW II: If the system is in the normalized state ψ, the outcome of an experiment that measures if it is in the normalized state φ is yes with probability and no with probability (1 p). p = φ ψ 2 (9) We stress that, if if the answer of this measurement is yes then, the system is in the state φ, i.e. the measurement affects the system. This is often called collapse of the wave function in QM books. It is worth mentioning that this interpretation makes sense only since p is in the range 0 p 1. This is true due the following mathematical result (known as Cauchy-Schwarz inequality): ψ φ ψ φ (10) for all ψ and φ. (This follows from the definition of a Hilbert space; I recommend that you review the proof of this result). 8. To formulate the next law we need more mathematical definitions: a self-adjoint operator 5  on a (finite dimensional 6 ) Hilbert space H is a linear map H H such that ( ψ ) = ψ Â. (11) If  obeys this condition one writes  = Â. Such self-adjoint operators have many nice mathematical properties described below. Before that I state why we are interested in such operators: 4 What we discussed in this paragraph is one aspect of QM that looks paradoxical to people used to the thinking of Newton s classical physics. To learn QM one needs to develop a new type of intuition, and this one can by carefully studying examples. 5 Here the self-adjoint means the same as hermitean, but this is not true in infinite dimensions. 6 In case of infinite dimensional Hilbert spaces the definition is somewhat more involved, but the condition below is still an essential property. 3
9. QM LAW III: To every physical property of the quantum system there exists an associated self-adjoint operator  on the Hilbert space H such that the expectation value of this physical property in an arbitrary state ψ is equal to  ψ = ψ A ψ ψ ψ. (12) One calls  ψ expectation value of the self-adjoint operator  in the state ψ. By physical property we mean something like the position, energy etc. that can be measured by an experiment. By expectation value we mean the average of the outcome of the same experiment: Suppose we make the following experiment on our system M 1 times: prepare the system in the state ψ and measure the physical property associated with Â. Then the outcome of this experiment can be different each time, and we therefore cannot predict it for sure. What we can predict is the average of the outcome: this average converges to  ψ for M. We can also make a prediction about how much the result will spread: consider the observable ˆB  A ψ. It measures how much the result of the experiment deviates from its most likely outcome. The expectation value of ˆB 2 is always nonnegative, and it is the larger the more the outcome of the individual experiments differ from their average. This latter expectation value is, by definition, the square of the so-called uncertainty ψ (A) of A, i.e. ψ (A) ( A ψ) 2 ψ = Â2 ψ  2 ψ. (13) Thus ψ (A) 0 is a good measure for how much the result of this experiment can be expected to spread. Note that ψ (A) = 0 if the outcome of the experiment described above can be predicted with certainty. Operators  representing physical properties are often called observables in the literature. I find this name somewhat misleading. However, since is has become a standard name in this context, I will also use it. 10. In our example all observables  corresponds to self-adjoint N N matrices (A jk ) N j,k=1 with A jk = A kj (14) for all j, k. Indeed, such a matrix defines a linear map that assigns states to other states according to the following rule:  ψ = ( A jk ψ k ) j, (15) j and the recipe in (15) exactly corresponds to matrix multiplication. Indeed, with the following correspondence k  A = (A jk ) N j,k=1 (16) 4
we have  ψ Aψ = A 11 A 12... A 1N A 21 A 22... A 2N...... ψ 1 ψ 2., (17) A N1 A N2... A NN ψ N i.e. the action of such an operator  just corresponds to matrix multiplication. Moreover, the condition in (14) is equivalent to the matrix A being self-adjoint. You should convince yourself that the condition in (14) is equivalent to the one (11). Warning: One often abuses notation and uses the same symbol H for an operator and the matrix representing it. We try to be clear here in this discussion by using different symbols Ĥ and H. However, later we will not always do that. 11. The main non-trivial mathematics result needed for these models is the Spectral theorem for self-adjoint matrices: Let A be a self-adjoint N N matrix. Then the eigenvalue equation Af = λf (18) (λ complex and f C N ) has N solutions λ j and f j such that all λ j are real and the f j are orthonormal: f j f k = δ jk (19) for all j, k = 1, 2,..., N. The f j and λ j are called eigenvectors and eigenstates of the self-adjoint matrix A. The difficult part of finite QM system problems is to compute the eigenvectors and eigenstates of a N N matrix. As you know, for N > 3 this is, in general, a difficult problem best solved numerically on a computer (using MATLAB or the like). However, this theorem has many interesting implications that are true for any model. For example: Let  be a self-adjoint operator. Then the theorem implies that there exist states f j and real numbers λ j such that (note this states are given by  f k = λ k f k, f j f k = δ jk. (20) f k = j (f k ) j j (21) with (f k ) j the components of the vector f k, which is just a special case of (1).) 5
12. Some remarks on how to use the rules above in practical computations: We can compute the action of  on any state ψ by expanding ψ as a linear combination of the states f j ; ψ = j f j f j ψ. (22) Indeed, (20) implies  ψ = j λ j f j f j ψ. (23) Moreover, for any function G(x) defined for real arguments x, we can define, G(Â) as follows, G(Â) ψ = G(λ j ) f j f j ψ. (24) j Examples of particular interest are the time evolution operator U(t) = e itĥ (t real) and the so-called resolvent H (a) = (z Ĥ) 1 of a Hamiltonian of the model. It therefore is natural to write G(Â) = j G(λ j ) f j f j. (25) 13. A set of states f j, j, k = 1, 2,... N, obeying the relations f j f k = δ jk is called orthonormal. The physical interpretation is as follows: The orthonormal eigenstates f j of the observable  correspond to states where  is a fixed value, and this value is given by the corresponding eigenvalue λ j. Indeed, this follows from QM LAW III above and f j Ân f j = λ n j (26) f j f j for n = 1, 2, implying  f j = λ j and fj (Â) = 0. Moreover, for a general normalized state ψ, we can compute the expectation value in (12) as A ψ = f j ψ 2 λ j. (27) j 14. Thus we have many other sets of special states f j where one can say, for sure, that some physical property of the system (i.e. the one corresponding to the observable Â) has one specific value. We now see that our orthonormal position states j are just one example out of many: we can construct an observable ˆX = j x j j j (28) with x j the value of the position in the state j. (At this point it is up to us to define the values of x j.) It is natural to call ˆX position operator. Our interpretation of the states j further above is then a special case of QM LAW III. 6
15. A particularly interested such observable is the Hamiltonian, usually denoted as Ĥ. It plays several roles in QM: Firstly, it is the energy operator, i.e. its eigenstates we denote them as E j are states where the energy has a fixed value we denote these corresponding values as E j. Secondly, the the Hamiltonian gives the time evolution of the system: QM LAW IV: If the system is at time t = 0 in the state ψ 0 and if the system is left alone (i.e. no experiment is performed on it) then, at time t > 0, it is in the state ψ(t) given by the solution of the following so-called Schrödinger equation: i ψ(t) = Ĥ ψ(t), (29) t with the initial condition ψ(0) = ψ ; is Planck s constant and i = 1. Note that ψ(t) is fixed by ψ(0), t, and Ĥ: the solution of (29) with the given initial condition is ψ(t) = e iĥt/ ψ 0 (30) (the proper interpretation of this was explained in the paragraph containing Eq. (25)). In this sense QM is deterministic. emark: For completeness we should mention that the Hamiltonian plays a 3rd role in quantum statistical physics: it also determines thermal equilibrium states. We will come back to this in a later lecture. 16. To sum up our discussion: A QM model is given by a Hilbert space H and a set of observables, i.e. selfadjoint operators, including the Hamiltonian Ĥ. To make physics predictions with such a model one needs to solve the Schrödinger equation and compute expectation values of observables. 17. I now give some more example of the usefulness of the Dirac bra-ket notation. In my discussion above I was careful to distinguish an abstract state ψ from its position representation ψ, i.e. the vector in C N with components ψ j = j ψ. This is important since one can use different representations as follows. Let b k be the eigenvalues of an observable ˆB and b k the corresponding orthonormal eigenstates: ˆB bk = b k b k. In this notation we write x j instead of j (since ˆX x j = x j x j, with the position operator ˆX introduced in (42)). We then can write, for any observable ˆB, ψ = k b k b k ψ (31) for states and  = j,k b j b j  b k b k = j,k b j  b k b j b k (32) 7
for observables: the vector in C N with the components b k ψ and the N N- matrix with the matrix elements b j A b k define the so-called ˆB-representation of the state ψ and the observable Â, respectively. The formulas above can be summarized by the following formula for the identity I: I = b k b k. (33) k Indeed, inserting this formula in ψ = I ψ and  = IÂI one gets the formulas in (31) and (32), respectively. How can one switch from the ˆX-representation to the ˆB-representation? All one needs to derive the needed equations is (33): x j ψ = k x j b k b k ψ (34) and x j  x k = m,n x j b m b m  b n b n x k (35) It is interesting to look in detail what the relation of this to a standard matrix notation is: One can regard the scalar products x j b m as elements of a N N matrix U: U jm = x j b m. Writing then ψ = ( b j ψ ) N j=1 (vector in C N ) and A = ( b j A b k ) N j,k=1 (N N- matrix) the above equations can be written as ψ = Uψ, A = UA U (36) with (U ) mj = x j b m = b m x j. Moreover, the relations in (33), and a similar relation for ˆX, imply UU = U U = I, (37) i.e. the matrix U is unitary. Thus the different representations of QM are related by unitary matrices. QM model of a particle moving on the real line 18. In this example the Hilbert space in the QM LAW I is L 2 (), i.e. the vector space of all complex valued functions ψ(x), x, such that ψ(x) 2 dx <, with the scalar product ψ φ = ψ(x)φ(x)dx. (38) 8
Observables of interest are the following: the position operator X, the momentum operator P, and the Hamiltonian H. They can be defined as follows, (Xψ)(x) = xψ(x) (P ψ)(x) = i d dx ψ(x) (Hψ)(x) = ( 2 d 2 ) 2m dx + V (x) ψ(x) 2 with m > 0 the mass parameter and a nice real-valued function V (x) corresponding to the potential. The time dependent Schrödinger equation in this case is a PDE: 7 i ( t ψ(x, t) = 2 2m d 2 ) dx + V (x) 2 (39) ψ(x, t). (40) In this example the Hilbert space is infinite dimensional, and therefore the mathematics is more complicated. However, many of the formal rules generalize in a somewhat natural manner if one uses the Dirac bra-ket notation. 19. The natural interpretation of ψ(x) is as follows: ψ(x) x ψ, (41) it is the pure state ψ in the position representation: there exist eigenstates x of the position operator ˆX labeled by the eigenvalues x : ˆX x = x x. (42) Moreover, these states are orthogonal in the following sense: x x = δ(x x ) x, x (43) with the Dirac delta δ(x x ), and these operators are complete in the following sense: dx x x = I. (44) Note the analogy of these formulas to the ones discussed above ψ j = x j ψ, ˆX x j = x j x j, x j x k = δ jk and k x k x k = I, respectively. In fact, it is possible to obtain from the latter equations the ones in (41) (44) above by a suitable limit N (this is discussed in [Shankar], Section 1.10). Thus (39) actually gives the representation of the position operator ˆX, momentum operator ˆP, and Hamiltonian Ĥ in position representation: (Xψ)(x) = x ˆX ψ etc. 7 partial differential equation 9
20. Another important representation is the momentum representation defined as follows: consider the eigenstates p of the momentum operator: together with the relations ˆP p = p p, p. (45) p p = δ(p p ), p p = I. (46) Then the state ψ in the momentum representation is defined as follows, ψ(p) p ψ. (47) What is the relation between the position- and the momentum representation? For that one needs to compute x p. From x ˆP p = p x p and one finds p p = dx p x x p = δ(p p ) x p = 1 2π e ipx/ = p x, (48) and with that ψ(p) = p ψ = dx p x x ψ = dx 1 e ipx/ ψ(x) 2π (49) 1 ψ(x) = x ψ = dp x p p ψ = dp e ipx/ ψ(p) 2π (check that!). Note that (49) is equivalent to the well-known formulas for Fourier transformations (to see that change variables to k = p/ and define ˆψ(k) = ψ( k)/ ). What are the momentum representations of the operators ˆX and ˆP? One finds X ψ(p) p ˆX ψ = i p ψ(p) P ψ(p) p ˆP ψ = p ψ(p) (50) (check that!). 21. What is the Hamiltonian Ĥ in momentum representation? By straightforward computations we find H ψ(p) p2 p Ĥ ψ = 2m ψ(p) + dp Ṽ (p p ) ψ(p ) (51) with Ṽ (p p ) = dx 2π e i(p p )x/ V (x) (52) (check if I got the signs right!). For V 0 this now is an integral operator! 10