(even if they are orthogonal), into its quantum operator is not at all straightforward. The mapping can always be done in terms of Cartesian coordinates after which a transformation of the resulting coordinates and differential operators to a curvilinear system can be performed The relationship of these quantum mechanical operators to experimental easurement lies in the eigenvalues of the quantum operators. Each such operator has a corresponding eigenvalue equa Fx=a;x in which the x, are called eigenfunctions and the(scalar numbers)a, are called eigenvalues. All such eigenvalue equations are posed in terms of a given operator(F in this case)and those functions (x,) that F acts on to produce the function back again but multiplied by a constant(the eigenvalue). Because the operator F usually contains differential operators(coming from the momentum), these equations are differential equations. Their solutions % depend on the coordinates that F contains as differential operators. An example will help clarify these points. The differential operator d/dy acts on what functions (of y) to generate the same function back again but multiplied by a constant? The answer is functions of the form exp(ay )since n)dy=a exp(ay) So, we say that exp(ay) is an eigenfunction of d/dy and a is the corresponding eigenvalue 21
21 (even if they are orthogonal), into its quantum operator is not at all straightforward. The mapping can always be done in terms of Cartesian coordinates after which a transformation of the resulting coordinates and differential operators to a curvilinear system can be performed. The relationship of these quantum mechanical operators to experimental measurement lies in the eigenvalues of the quantum operators. Each such operator has a corresponding eigenvalue equation F cj = aj cj in which the cj are called eigenfunctions and the (scalar numbers) aj are called eigenvalues. All such eigenvalue equations are posed in terms of a given operator (F in this case) and those functions {cj} that F acts on to produce the function back again but multiplied by a constant (the eigenvalue). Because the operator F usually contains differential operators (coming from the momentum), these equations are differential equations. Their solutions cj depend on the coordinates that F contains as differential operators. An example will help clarify these points. The differential operator d/dy acts on what functions (of y) to generate the same function back again but multiplied by a constant? The answer is functions of the form exp(ay) since d (exp(ay))/dy = a exp(ay). So, we say that exp(ay) is an eigenfunction of d/dy and a is the corresponding eigenvalue
As I will discuss in more detail shortly, the eigenvalues of the operator F tell us the only values of the physical property corresponding to the operator F that can be observed in a laboratory measurement. Some F operators that we encounter pos eigenvalues that are discrete or quantized. For such properties, laboratory measurement will result in only those discrete values. Other F operators have eigenvalues that can tak on a continuous range of values; for these properties, laboratory measurement can give any value in this continuous range B. Wave functions The eigenfunctions of a quantum mechanical operator depend on the coordinate upon which the operator acts. The particular operator that corresponds to the total energy of the system is called the Hamiltonian operator:. The eigenfunctions of this particular operator are called wave functions A special case of an operator corresponding to a physically measurable quantity is the Hamiltonian operator h that relates to the total energy of the system The energy eigenstates of the system y are functions of the coordinates qi that H depends on and of time t. The function/ Y(q;, t)/2=Y*y gives the probability density for observing the coordinates at the values qi at time t For a many-particle system such as the H2O molecule, the wave function depends on many coordinates. For H2O, it depends on the x y, and z(or r, e, and o) coordinates of the ten electrons and the x, y, and z(or r, e, and o) coordinates of the oxygen nucleus and of the two protons; a total of thirty-nine coordinates appear in p
22 As I will discuss in more detail shortly, the eigenvalues of the operator F tell us the only values of the physical property corresponding to the operator F that can be observed in a laboratory measurement. Some F operators that we encounter possess eigenvalues that are discrete or quantized. For such properties, laboratory measurement will result in only those discrete values. Other F operators have eigenvalues that can take on a continuous range of values; for these properties, laboratory measurement can give any value in this continuous range. B. Wave functions The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts. The particular operator that corresponds to the total energy of the system is called the Hamiltonian operator. The eigenfunctions of this particular operator are called wave functions A special case of an operator corresponding to a physically measurable quantity is the Hamiltonian operator H that relates to the total energy of the system. The energy eigenstates of the system Y are functions of the coordinates {qj} that H depends on and of time t. The function |Y(qj ,t)|2 = Y*Y gives the probability density for observing the coordinates at the values qj at time t. For a many-particle system such as the H2O molecule, the wave function depends on many coordinates. For H2O, it depends on the x, y, and z (or r,q, and f) coordinates of the ten electrons and the x, y, and z (or r,q, and f) coordinates of the oxygen nucleus and of the two protons; a total of thirty-nine coordinates appear in Y
In classical mechanics, the coordinates qj and their corresponding momenta p are functions of time. The state of the system is then described by specifying qi(t)and pi (t) In quantum mechanics, the concept that qi is known as a function of time is replaced by the concept of the probability density for finding qj at a particular value at a particular time y(qi, t)2. Knowledge of the corresponding momenta as functions of time is also relinquished in quantum mechanics; again, only knowledge of the probability density for finding P] with any particular value at a particular time t remains The hamiltonian eigenstates are especially important in chemistry because many of the tools that chemists use to study molecules probe the energy states of the molecule For example, most spectroscopic methods are designed to determine which energy state a molecule is in. However, there are other experimental measurements that measure other properties(e.g, the z-component of angular momentum or the total angular momentum) As stated earlier, if the state of some molecular system is characterized by a wave function p that happens to be an eigenfunction of a quantum mechanical operator F,one can immediately say something about what the outcome will be if the physical property F corresponding to the operator F is measured. In particular, since x=nixi where n, is one of the eigenvalues of F, we know that the value A; will be observed if the property F is measured while the molecule is described by the wave function =%. In fact, once a measurement of a physical quantity F has been carried out and a particular eigenvalue 7i has been observed, the systems wave function Ybe
23 In classical mechanics, the coordinates qj and their corresponding momenta pj are functions of time. The state of the system is then described by specifying qj (t) and pj (t). In quantum mechanics, the concept that qj is known as a function of time is replaced by the concept of the probability density for finding qj at a particular value at a particular time |Y(qj ,t)|2. Knowledge of the corresponding momenta as functions of time is also relinquished in quantum mechanics; again, only knowledge of the probability density for finding pj with any particular value at a particular time t remains. The Hamiltonian eigenstates are especially important in chemistry because many of the tools that chemists use to study molecules probe the energy states of the molecule. For example, most spectroscopic methods are designed to determine which energy state a molecule is in. However, there are other experimental measurements that measure other properties (e.g., the z-component of angular momentum or the total angular momentum). As stated earlier, if the state of some molecular system is characterized by a wave function Y that happens to be an eigenfunction of a quantum mechanical operator F, one can immediately say something about what the outcome will be if the physical property F corresponding to the operator F is measured. In particular, since F cj = lj cj , where lj is one of the eigenvalues of F, we know that the value lj will be observed if the property F is measured while the molecule is described by the wave function Y = cj . In fact, once a measurement of a physical quantity F has been carried out and a particular eigenvalue lj has been observed, the system's wave function Y becomes the
eigenfunction % that corresponds to that eigenvalue. That is, the act of making the measurement causes the system's wave function to become the eigenfunction of the property that was measured What happens if some other property G, whose quantum mechanical operator is G is measured in such a case? We know from what was said earlier that some eigenvalue Hk of the operator g will be observed in the measurement. But, will the molecule's wave function remain. after G is measured. the eigenfunction of f or will the measurement of G cause p to be altered in a way that makes the molecule's state no longer an eigenfunction of F? It turns out that if the two operators f and g obey the condition FGEGF then, when the property G is measured, the wave function Y =%i will remain unchanged This property that the order of application of the two operators does not matter is called commutation; that is, we say the two operators commute if they obey this property. Let us see how this property leads to the conclusion about y remaining unchanged if the two operators commute. In particular, we apply the g operator to the above eigenval equation GFx=G入x Next, we use the commutation to re-write the left-hand side of this equation, and use the fact that 2 is a scalar number to thus obtain 4
24 eigenfunction cj that corresponds to that eigenvalue. That is, the act of making the measurement causes the system's wave function to become the eigenfunction of the property that was measured. What happens if some other property G, whose quantum mechanical operator is G is measured in such a case? We know from what was said earlier that some eigenvalue mk of the operator G will be observed in the measurement. But, will the molecule's wave function remain, after G is measured, the eigenfunction of F, or will the measurement of G cause Y to be altered in a way that makes the molecule's state no longer an eigenfunction of F? It turns out that if the two operators F and G obey the condition F G = G F, then, when the property G is measured, the wave function Y = cj will remain unchanged. This property that the order of application of the two operators does not matter is called commutation; that is, we say the two operators commute if they obey this property. Let us see how this property leads to the conclusion about Y remaining unchanged if the two operators commute. In particular, we apply the G operator to the above eigenvalue equation: G F cj = G lj cj . Next, we use the commutation to re-write the left-hand side of this equation, and use the fact that lj is a scalar number to thus obtain:
FGx=, GX So, now we see that(G)itself is an eigenfunction of F having eigenvalue M,. So, unless there are more than one eigenfunction of F corresponding to the eigenvalue m(i.e, unless this eigenvalue is degenerate), Gx, must itself be proportional to x. We write this proportionality conclusion as G XiNji which means that ti is also an eigenfunction of G. This, in turn, means that measuring the property g while the system is described by the wave function p =%i does not change the wave function; it remains x; So, when the operators corresponding to two physical properties commute, once one measures one of the properties(and thus causes the system to be an eigenfunction of that operator), subsequent measurement of the second operator will (if the eigenvalue of the first operator is not degenerate) produce a unique eigenvalue of the second operator and will not change the system wave function If the two operators do not commute, one simply can not reach the above conclusions. In such cases, measurement of the property corresponding to the first operator will lead to one of the eigenvalues of that operator and cause the system wave function to become the corresponding eigenfuction. However, subsequent measurement of the second operator will produce an eigenvalue of that operator, but the system wave
25 F G cj = lj Gcj . So, now we see that (Gcj ) itself is an eigenfunction of F having eigenvalue lj . So, unless there are more than one eigenfunction of F corresponding to the eigenvalue lj (i.e., unless this eigenvalue is degenerate), Gcj must itself be proportional to cj . We write this proportionality conclusion as G cj = mj cj , which means that cj is also an eigenfunction of G. This, in turn, means that measuring the property G while the system is described by the wave function Y = cj does not change the wave function; it remains cj . So, when the operators corresponding to two physical properties commute, once one measures one of the properties (and thus causes the system to be an eigenfunction of that operator), subsequent measurement of the second operator will (if the eigenvalue of the first operator is not degenerate) produce a unique eigenvalue of the second operator and will not change the system wave function. If the two operators do not commute, one simply can not reach the above conclusions. In such cases, measurement of the property corresponding to the first operator will lead to one of the eigenvalues of that operator and cause the system wave function to become the corresponding eigenfuction. However, subsequent measurement of the second operator will produce an eigenvalue of that operator, but the system wave