Page 1 Chapter 6 The Schr¨ odinger Wave Equation So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a â€˜wave equationâ€™ for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will o?er a simple derivation based on what we have learned so far about the wave function. The Schr¨ odinger equation has two â€˜formsâ€™, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schr¨ odinger wave equation. The other is the equation in which the time dependence has been â€˜removedâ€™ and hence is known as the time independent Schr¨ odinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations â€“ the time independent equation can be derived readily from the time dependent equation (except if the potential is time dependent, a development we will not be discussing here). In the following we will describe how the ?rst, time dependent equation can be â€˜derivedâ€™, and in then how the second follows from the ?rst. 6.1 Derivation of the Schr¨ odinger Wave Equation 6.1.1 The Time Dependent Schr¨ odinger Wave Equation In the discussion of the particle in an in?nite potential well, it was observed that the wave function of a particle of ?xed energy E could most naturally be written as a linear combination of wave functions of the form ?(x,t) = Ae i(kx-?t) (6.1) representing a wave travelling in the positive x direction, and a corresponding wave trav- elling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function Page 2 Chapter 6 The Schr¨ odinger Wave Equation So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a â€˜wave equationâ€™ for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will o?er a simple derivation based on what we have learned so far about the wave function. The Schr¨ odinger equation has two â€˜formsâ€™, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schr¨ odinger wave equation. The other is the equation in which the time dependence has been â€˜removedâ€™ and hence is known as the time independent Schr¨ odinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations â€“ the time independent equation can be derived readily from the time dependent equation (except if the potential is time dependent, a development we will not be discussing here). In the following we will describe how the ?rst, time dependent equation can be â€˜derivedâ€™, and in then how the second follows from the ?rst. 6.1 Derivation of the Schr¨ odinger Wave Equation 6.1.1 The Time Dependent Schr¨ odinger Wave Equation In the discussion of the particle in an in?nite potential well, it was observed that the wave function of a particle of ?xed energy E could most naturally be written as a linear combination of wave functions of the form ?(x,t) = Ae i(kx-?t) (6.1) representing a wave travelling in the positive x direction, and a corresponding wave trav- elling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function Chapter 6 The Schr¨ odinger Wave Equation 43 for a free particle of momentum p = !k and energy E = !?. With this in mind, we can then note that ? 2 ? ?x 2 =-k 2 ? (6.2) which can be written, using E = p 2 /2m = ! 2 k 2 /2m: - ! 2 2m ? 2 ? ?x 2 = p 2 2m ?. (6.3) Similarly ?? ?t =-i?? (6.4) which can be written, using E = !?: i! ?? ?t = !?? = E?. (6.5) We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then E = p 2 /2m + V (x) so that E? = p 2 2m ? + V (x)? (6.6) where? is now the wave function of a particle moving in the presence of a potential V (x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have - ! 2 2m ? 2 ? ?x 2 + V (x)? = i! ?? ?t (6.7) which is the famous time dependent Schr¨ odinger wave equation. It is setting up and solving this equation, then analyzing the physical contents of its solutions that form the basis of that branch of quantum mechanics known as wave mechanics. Even though this equation does not look like the familiar wave equation that describes, for instance, waves on a stretched string, it is nevertheless referred to as a â€˜wave equationâ€™ as it can have solutions that represent waves propagating through space. We have seen an example of this: the harmonic wave function for a free particle of energy E and momentum p, i.e. ?(x,t) = Ae -i(px-Et)/! (6.8) is a solution of this equation with, as appropriate for a free particle, V (x) = 0. But this equation can have distinctly non-wave like solutions whose form depends, amongst other things, on the nature of the potential V (x) experienced by the particle. In general, the solutions to the time dependent Schr¨ odinger equation will describe the dynamical behaviour of the particle, in some sense similar to the way that Newtonâ€™s equation F = ma describes the dynamics of a particle in classical physics. However, there is an important di?erence. By solving Newtonâ€™s equation we can determine the position of a particle as a function of time, whereas by solving Schr¨ odingerâ€™s equation, what we get is a wave function?(x,t) which tells us (after we square the wave function) how the probability of ?nding the particle in some region in space varies as a function of time. It is possible to proceed from here look at ways and means of solving the full, time dependent Schr¨ odinger equation in all its glory, and look for the physical meaning of the solutions that are found. However this route, in a sense, bypasses much important physics contained in the Schr¨ odinger equation which we can get at by asking much simpler questions. Perhaps the most important â€˜simpler questionâ€™ to ask is this: what is the wave Page 3 Chapter 6 The Schr¨ odinger Wave Equation So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a â€˜wave equationâ€™ for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will o?er a simple derivation based on what we have learned so far about the wave function. The Schr¨ odinger equation has two â€˜formsâ€™, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schr¨ odinger wave equation. The other is the equation in which the time dependence has been â€˜removedâ€™ and hence is known as the time independent Schr¨ odinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations â€“ the time independent equation can be derived readily from the time dependent equation (except if the potential is time dependent, a development we will not be discussing here). In the following we will describe how the ?rst, time dependent equation can be â€˜derivedâ€™, and in then how the second follows from the ?rst. 6.1 Derivation of the Schr¨ odinger Wave Equation 6.1.1 The Time Dependent Schr¨ odinger Wave Equation In the discussion of the particle in an in?nite potential well, it was observed that the wave function of a particle of ?xed energy E could most naturally be written as a linear combination of wave functions of the form ?(x,t) = Ae i(kx-?t) (6.1) representing a wave travelling in the positive x direction, and a corresponding wave trav- elling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function Chapter 6 The Schr¨ odinger Wave Equation 43 for a free particle of momentum p = !k and energy E = !?. With this in mind, we can then note that ? 2 ? ?x 2 =-k 2 ? (6.2) which can be written, using E = p 2 /2m = ! 2 k 2 /2m: - ! 2 2m ? 2 ? ?x 2 = p 2 2m ?. (6.3) Similarly ?? ?t =-i?? (6.4) which can be written, using E = !?: i! ?? ?t = !?? = E?. (6.5) We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then E = p 2 /2m + V (x) so that E? = p 2 2m ? + V (x)? (6.6) where? is now the wave function of a particle moving in the presence of a potential V (x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have - ! 2 2m ? 2 ? ?x 2 + V (x)? = i! ?? ?t (6.7) which is the famous time dependent Schr¨ odinger wave equation. It is setting up and solving this equation, then analyzing the physical contents of its solutions that form the basis of that branch of quantum mechanics known as wave mechanics. Even though this equation does not look like the familiar wave equation that describes, for instance, waves on a stretched string, it is nevertheless referred to as a â€˜wave equationâ€™ as it can have solutions that represent waves propagating through space. We have seen an example of this: the harmonic wave function for a free particle of energy E and momentum p, i.e. ?(x,t) = Ae -i(px-Et)/! (6.8) is a solution of this equation with, as appropriate for a free particle, V (x) = 0. But this equation can have distinctly non-wave like solutions whose form depends, amongst other things, on the nature of the potential V (x) experienced by the particle. In general, the solutions to the time dependent Schr¨ odinger equation will describe the dynamical behaviour of the particle, in some sense similar to the way that Newtonâ€™s equation F = ma describes the dynamics of a particle in classical physics. However, there is an important di?erence. By solving Newtonâ€™s equation we can determine the position of a particle as a function of time, whereas by solving Schr¨ odingerâ€™s equation, what we get is a wave function?(x,t) which tells us (after we square the wave function) how the probability of ?nding the particle in some region in space varies as a function of time. It is possible to proceed from here look at ways and means of solving the full, time dependent Schr¨ odinger equation in all its glory, and look for the physical meaning of the solutions that are found. However this route, in a sense, bypasses much important physics contained in the Schr¨ odinger equation which we can get at by asking much simpler questions. Perhaps the most important â€˜simpler questionâ€™ to ask is this: what is the wave Chapter 6 The Schr¨ odinger Wave Equation 44 function for a particle of a given energy E? Curiously enough, to answer this question requires â€˜extractingâ€™ the time dependence from the time dependent Schr¨ odinger equation. To see how this is done, and its consequences, we will turn our attention to the closely related time independent version of this equation. 6.1.2 The Time Independent Schr¨ odinger Equation We have seen what the wave function looks like for a free particle of energy E â€“ one or the other of the harmonic wave functions â€“ and we have seen what it looks like for the particle in an in?nitely deep potential well â€“ see Section 5.3 â€“ though we did not obtain that result by solving the Schr¨ odinger equation. But in both cases, the time dependence entered into the wave function via a complex exponential factor exp[-iEt/!]. This suggests that to â€˜extractâ€™ this time dependence we guess a solution to the Schr¨ odinger wave equation of the form ?(x,t) =?(x)e -iEt/! (6.9) i.e. where the space and the time dependence of the complete wave function are contained in separate factors 1 . The idea now is to see if this guess enables us to derive an equation for ?(x), the spatial part of the wave function. If we substitute this trial solution into the Schr¨ odinger wave equation, and make use of the meaning of partial derivatives, we get: - ! 2 2m d 2 ?(x) dx 2 e -iEt/! +V (x)?(x)e -iEt/! = i!.-iE/!e -iEt/! ?(x) = E?(x)e -iEt/! . (6.10) We now see that the factor exp[-iEt/!] cancels from both sides of the equation, giving us - ! 2 2m d 2 ?(x) dx 2 + V (x)?(x) = E?(x) (6.11) If we rearrange the terms, we end up with ! 2 2m d 2 ?(x) dx 2 + ! E- V (x) " ?(x) = 0 (6.12) which is the time independent Schr¨ odinger equation. We note here that the quantity E, which we have identi?ed as the energy of the particle, is a free parameter in this equation. In other words, at no stage has any restriction been placed on the possible values for E. Thus, if we want to determine the wave function for a particle with some speci?c value of E that is moving in the presence of a potential V (x), all we have to do is to insert this value of E into the equation with the appropriate V (x), and solve for the corresponding wave function. In doing so, we ?nd, perhaps not surprisingly, that for di?erent choices of E we get di?erent solutions for?(x). We can emphasize this fact by writing? E (x) as the solution associated with a particular value of E. But it turns out that it is not all quite as simple as this. To be physically acceptable, the wave function? E (x) must satisfy two conditions, one of which we have seen before namely that the wave function must be normalizable (see Eq. (5.3)), and a second, that the wave function and its derivative must be continuous. Together, these two requirements, the ?rst founded in the probability interpretation of the wave function, the second in more esoteric mathematical necessities which we will not go into here and usually only encountered in somewhat arti?cial problems, lead to a rather remarkable property of physical systems described by this equation that has enormous physical signi?cance: the quantization of energy. 1 A solution of this form can be shown to arise by the method of â€˜the separation of variablesâ€™, a well known mathematical technique used to solve equations of the form of the Schr¨ odinger equation. Page 4 Chapter 6 The Schr¨ odinger Wave Equation So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a â€˜wave equationâ€™ for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will o?er a simple derivation based on what we have learned so far about the wave function. The Schr¨ odinger equation has two â€˜formsâ€™, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schr¨ odinger wave equation. The other is the equation in which the time dependence has been â€˜removedâ€™ and hence is known as the time independent Schr¨ odinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations â€“ the time independent equation can be derived readily from the time dependent equation (except if the potential is time dependent, a development we will not be discussing here). In the following we will describe how the ?rst, time dependent equation can be â€˜derivedâ€™, and in then how the second follows from the ?rst. 6.1 Derivation of the Schr¨ odinger Wave Equation 6.1.1 The Time Dependent Schr¨ odinger Wave Equation In the discussion of the particle in an in?nite potential well, it was observed that the wave function of a particle of ?xed energy E could most naturally be written as a linear combination of wave functions of the form ?(x,t) = Ae i(kx-?t) (6.1) representing a wave travelling in the positive x direction, and a corresponding wave trav- elling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function Chapter 6 The Schr¨ odinger Wave Equation 43 for a free particle of momentum p = !k and energy E = !?. With this in mind, we can then note that ? 2 ? ?x 2 =-k 2 ? (6.2) which can be written, using E = p 2 /2m = ! 2 k 2 /2m: - ! 2 2m ? 2 ? ?x 2 = p 2 2m ?. (6.3) Similarly ?? ?t =-i?? (6.4) which can be written, using E = !?: i! ?? ?t = !?? = E?. (6.5) We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then E = p 2 /2m + V (x) so that E? = p 2 2m ? + V (x)? (6.6) where? is now the wave function of a particle moving in the presence of a potential V (x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have - ! 2 2m ? 2 ? ?x 2 + V (x)? = i! ?? ?t (6.7) which is the famous time dependent Schr¨ odinger wave equation. It is setting up and solving this equation, then analyzing the physical contents of its solutions that form the basis of that branch of quantum mechanics known as wave mechanics. Even though this equation does not look like the familiar wave equation that describes, for instance, waves on a stretched string, it is nevertheless referred to as a â€˜wave equationâ€™ as it can have solutions that represent waves propagating through space. We have seen an example of this: the harmonic wave function for a free particle of energy E and momentum p, i.e. ?(x,t) = Ae -i(px-Et)/! (6.8) is a solution of this equation with, as appropriate for a free particle, V (x) = 0. But this equation can have distinctly non-wave like solutions whose form depends, amongst other things, on the nature of the potential V (x) experienced by the particle. In general, the solutions to the time dependent Schr¨ odinger equation will describe the dynamical behaviour of the particle, in some sense similar to the way that Newtonâ€™s equation F = ma describes the dynamics of a particle in classical physics. However, there is an important di?erence. By solving Newtonâ€™s equation we can determine the position of a particle as a function of time, whereas by solving Schr¨ odingerâ€™s equation, what we get is a wave function?(x,t) which tells us (after we square the wave function) how the probability of ?nding the particle in some region in space varies as a function of time. It is possible to proceed from here look at ways and means of solving the full, time dependent Schr¨ odinger equation in all its glory, and look for the physical meaning of the solutions that are found. However this route, in a sense, bypasses much important physics contained in the Schr¨ odinger equation which we can get at by asking much simpler questions. Perhaps the most important â€˜simpler questionâ€™ to ask is this: what is the wave Chapter 6 The Schr¨ odinger Wave Equation 44 function for a particle of a given energy E? Curiously enough, to answer this question requires â€˜extractingâ€™ the time dependence from the time dependent Schr¨ odinger equation. To see how this is done, and its consequences, we will turn our attention to the closely related time independent version of this equation. 6.1.2 The Time Independent Schr¨ odinger Equation We have seen what the wave function looks like for a free particle of energy E â€“ one or the other of the harmonic wave functions â€“ and we have seen what it looks like for the particle in an in?nitely deep potential well â€“ see Section 5.3 â€“ though we did not obtain that result by solving the Schr¨ odinger equation. But in both cases, the time dependence entered into the wave function via a complex exponential factor exp[-iEt/!]. This suggests that to â€˜extractâ€™ this time dependence we guess a solution to the Schr¨ odinger wave equation of the form ?(x,t) =?(x)e -iEt/! (6.9) i.e. where the space and the time dependence of the complete wave function are contained in separate factors 1 . The idea now is to see if this guess enables us to derive an equation for ?(x), the spatial part of the wave function. If we substitute this trial solution into the Schr¨ odinger wave equation, and make use of the meaning of partial derivatives, we get: - ! 2 2m d 2 ?(x) dx 2 e -iEt/! +V (x)?(x)e -iEt/! = i!.-iE/!e -iEt/! ?(x) = E?(x)e -iEt/! . (6.10) We now see that the factor exp[-iEt/!] cancels from both sides of the equation, giving us - ! 2 2m d 2 ?(x) dx 2 + V (x)?(x) = E?(x) (6.11) If we rearrange the terms, we end up with ! 2 2m d 2 ?(x) dx 2 + ! E- V (x) " ?(x) = 0 (6.12) which is the time independent Schr¨ odinger equation. We note here that the quantity E, which we have identi?ed as the energy of the particle, is a free parameter in this equation. In other words, at no stage has any restriction been placed on the possible values for E. Thus, if we want to determine the wave function for a particle with some speci?c value of E that is moving in the presence of a potential V (x), all we have to do is to insert this value of E into the equation with the appropriate V (x), and solve for the corresponding wave function. In doing so, we ?nd, perhaps not surprisingly, that for di?erent choices of E we get di?erent solutions for?(x). We can emphasize this fact by writing? E (x) as the solution associated with a particular value of E. But it turns out that it is not all quite as simple as this. To be physically acceptable, the wave function? E (x) must satisfy two conditions, one of which we have seen before namely that the wave function must be normalizable (see Eq. (5.3)), and a second, that the wave function and its derivative must be continuous. Together, these two requirements, the ?rst founded in the probability interpretation of the wave function, the second in more esoteric mathematical necessities which we will not go into here and usually only encountered in somewhat arti?cial problems, lead to a rather remarkable property of physical systems described by this equation that has enormous physical signi?cance: the quantization of energy. 1 A solution of this form can be shown to arise by the method of â€˜the separation of variablesâ€™, a well known mathematical technique used to solve equations of the form of the Schr¨ odinger equation. Chapter 6 The Schr¨ odinger Wave Equation 45 The Quantization of Energy At ?rst thought it might seem to be perfectly acceptable to insert any value of E into the time independent Schr¨ odinger equation and solve it for ? E (x). But in doing so we must remain aware of one further requirement of a wave function which comes from its probability interpretation: to be physically acceptable a wave function must satisfy the normalization condition, Eq. (5.3) # +8 -8 |?(x,t)| 2 dx = 1 for all time t. For the particular trial solution introduced above, Eq. (6.9): ?(x,t) =? E (x)e -iEt/! (6.13) the requirement that the normalization condition must hold gives, on substituting for ?(x,t), the result 2 # +8 -8 |?(x,t)| 2 dx = # +8 -8 |? E (x)| 2 dx = 1. (6.14) Since this integral must be ?nite, (unity in fact), we must have ? E (x)? 0 as x? ±8 in order for the integral to have any hope of converging to a ?nite value. The importance of this with regard to solving the time dependent Schr¨ odinger equation is that we must check whether or not a solution ? E (x) obtained for some chosen value of E satis?es the normalization condition. If it does, then this is a physically acceptable solution, if it does not, then that solution and the corresponding value of the energy are not physically acceptable. The particular case of considerable physical signi?cance is if the potential V (x) is attractive, such as would be found with an electron caught by the attractive Coulomb force of an atomic nucleus, or a particle bound by a simple harmonic potential (a mass on a spring), or, as we have seen in Section 5.3, a particle trapped in an in?nite potential well. In all such cases, we ?nd that except for certain discrete values of the energy, the wave function ? E (x) does not vanish, or even worse, diverges, as x? ±8. In other words, it is only for these discrete values of the energy E that we get physically acceptable wave functions ? E (x), or to put it more bluntly, the particle can never be observed to have any energy other than these particular values, for which reason these energies are often referred to as the â€˜allowedâ€™ energies of the particle. This pairing o? of allowed energy and normalizable wave function is referred to mathematically as ? E (x) being an eigenfunction of the Schr¨ odinger equation, and E the associated energy eigenvalue, a terminology that acquires more meaning when quantum mechanics is looked at from a more advanced standpoint. So we have the amazing result that the probability interpretation of the wave function forces us to conclude that the allowed energies of a particle moving in a potential V (x) are restricted to certain discrete values, these values determined by the nature of the po- tential. This is the phenomenon known as the quantization of energy, a result of quantum mechanics which has enormous signi?cance for determining the structure of atoms, or, to go even further, the properties of matter overall. We have already seen an example of this quantization of energy in our earlier discussion of a particle in an in?ntely deep potential 2 Note that the time dependence has cancelled out because |?(x,t)| 2 = |?E(x)e -iEt/! | 2 = |?E(x)| 2 |e -iEt/! | 2 = |?E(x)| 2 since, for any complex number of the form exp(if), we have |exp(if)| 2 = 1. Page 5 Chapter 6 The Schr¨ odinger Wave Equation So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a â€˜wave equationâ€™ for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will o?er a simple derivation based on what we have learned so far about the wave function. The Schr¨ odinger equation has two â€˜formsâ€™, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. In general, the wave function behaves like a wave, and so the equation is often referred to as the time dependent Schr¨ odinger wave equation. The other is the equation in which the time dependence has been â€˜removedâ€™ and hence is known as the time independent Schr¨ odinger equation and is found to describe, amongst other things, what the allowed energies are of the particle. These are not two separate, independent equations â€“ the time independent equation can be derived readily from the time dependent equation (except if the potential is time dependent, a development we will not be discussing here). In the following we will describe how the ?rst, time dependent equation can be â€˜derivedâ€™, and in then how the second follows from the ?rst. 6.1 Derivation of the Schr¨ odinger Wave Equation 6.1.1 The Time Dependent Schr¨ odinger Wave Equation In the discussion of the particle in an in?nite potential well, it was observed that the wave function of a particle of ?xed energy E could most naturally be written as a linear combination of wave functions of the form ?(x,t) = Ae i(kx-?t) (6.1) representing a wave travelling in the positive x direction, and a corresponding wave trav- elling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function Chapter 6 The Schr¨ odinger Wave Equation 43 for a free particle of momentum p = !k and energy E = !?. With this in mind, we can then note that ? 2 ? ?x 2 =-k 2 ? (6.2) which can be written, using E = p 2 /2m = ! 2 k 2 /2m: - ! 2 2m ? 2 ? ?x 2 = p 2 2m ?. (6.3) Similarly ?? ?t =-i?? (6.4) which can be written, using E = !?: i! ?? ?t = !?? = E?. (6.5) We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then E = p 2 /2m + V (x) so that E? = p 2 2m ? + V (x)? (6.6) where? is now the wave function of a particle moving in the presence of a potential V (x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have - ! 2 2m ? 2 ? ?x 2 + V (x)? = i! ?? ?t (6.7) which is the famous time dependent Schr¨ odinger wave equation. It is setting up and solving this equation, then analyzing the physical contents of its solutions that form the basis of that branch of quantum mechanics known as wave mechanics. Even though this equation does not look like the familiar wave equation that describes, for instance, waves on a stretched string, it is nevertheless referred to as a â€˜wave equationâ€™ as it can have solutions that represent waves propagating through space. We have seen an example of this: the harmonic wave function for a free particle of energy E and momentum p, i.e. ?(x,t) = Ae -i(px-Et)/! (6.8) is a solution of this equation with, as appropriate for a free particle, V (x) = 0. But this equation can have distinctly non-wave like solutions whose form depends, amongst other things, on the nature of the potential V (x) experienced by the particle. In general, the solutions to the time dependent Schr¨ odinger equation will describe the dynamical behaviour of the particle, in some sense similar to the way that Newtonâ€™s equation F = ma describes the dynamics of a particle in classical physics. However, there is an important di?erence. By solving Newtonâ€™s equation we can determine the position of a particle as a function of time, whereas by solving Schr¨ odingerâ€™s equation, what we get is a wave function?(x,t) which tells us (after we square the wave function) how the probability of ?nding the particle in some region in space varies as a function of time. It is possible to proceed from here look at ways and means of solving the full, time dependent Schr¨ odinger equation in all its glory, and look for the physical meaning of the solutions that are found. However this route, in a sense, bypasses much important physics contained in the Schr¨ odinger equation which we can get at by asking much simpler questions. Perhaps the most important â€˜simpler questionâ€™ to ask is this: what is the wave Chapter 6 The Schr¨ odinger Wave Equation 44 function for a particle of a given energy E? Curiously enough, to answer this question requires â€˜extractingâ€™ the time dependence from the time dependent Schr¨ odinger equation. To see how this is done, and its consequences, we will turn our attention to the closely related time independent version of this equation. 6.1.2 The Time Independent Schr¨ odinger Equation We have seen what the wave function looks like for a free particle of energy E â€“ one or the other of the harmonic wave functions â€“ and we have seen what it looks like for the particle in an in?nitely deep potential well â€“ see Section 5.3 â€“ though we did not obtain that result by solving the Schr¨ odinger equation. But in both cases, the time dependence entered into the wave function via a complex exponential factor exp[-iEt/!]. This suggests that to â€˜extractâ€™ this time dependence we guess a solution to the Schr¨ odinger wave equation of the form ?(x,t) =?(x)e -iEt/! (6.9) i.e. where the space and the time dependence of the complete wave function are contained in separate factors 1 . The idea now is to see if this guess enables us to derive an equation for ?(x), the spatial part of the wave function. If we substitute this trial solution into the Schr¨ odinger wave equation, and make use of the meaning of partial derivatives, we get: - ! 2 2m d 2 ?(x) dx 2 e -iEt/! +V (x)?(x)e -iEt/! = i!.-iE/!e -iEt/! ?(x) = E?(x)e -iEt/! . (6.10) We now see that the factor exp[-iEt/!] cancels from both sides of the equation, giving us - ! 2 2m d 2 ?(x) dx 2 + V (x)?(x) = E?(x) (6.11) If we rearrange the terms, we end up with ! 2 2m d 2 ?(x) dx 2 + ! E- V (x) " ?(x) = 0 (6.12) which is the time independent Schr¨ odinger equation. We note here that the quantity E, which we have identi?ed as the energy of the particle, is a free parameter in this equation. In other words, at no stage has any restriction been placed on the possible values for E. Thus, if we want to determine the wave function for a particle with some speci?c value of E that is moving in the presence of a potential V (x), all we have to do is to insert this value of E into the equation with the appropriate V (x), and solve for the corresponding wave function. In doing so, we ?nd, perhaps not surprisingly, that for di?erent choices of E we get di?erent solutions for?(x). We can emphasize this fact by writing? E (x) as the solution associated with a particular value of E. But it turns out that it is not all quite as simple as this. To be physically acceptable, the wave function? E (x) must satisfy two conditions, one of which we have seen before namely that the wave function must be normalizable (see Eq. (5.3)), and a second, that the wave function and its derivative must be continuous. Together, these two requirements, the ?rst founded in the probability interpretation of the wave function, the second in more esoteric mathematical necessities which we will not go into here and usually only encountered in somewhat arti?cial problems, lead to a rather remarkable property of physical systems described by this equation that has enormous physical signi?cance: the quantization of energy. 1 A solution of this form can be shown to arise by the method of â€˜the separation of variablesâ€™, a well known mathematical technique used to solve equations of the form of the Schr¨ odinger equation. Chapter 6 The Schr¨ odinger Wave Equation 45 The Quantization of Energy At ?rst thought it might seem to be perfectly acceptable to insert any value of E into the time independent Schr¨ odinger equation and solve it for ? E (x). But in doing so we must remain aware of one further requirement of a wave function which comes from its probability interpretation: to be physically acceptable a wave function must satisfy the normalization condition, Eq. (5.3) # +8 -8 |?(x,t)| 2 dx = 1 for all time t. For the particular trial solution introduced above, Eq. (6.9): ?(x,t) =? E (x)e -iEt/! (6.13) the requirement that the normalization condition must hold gives, on substituting for ?(x,t), the result 2 # +8 -8 |?(x,t)| 2 dx = # +8 -8 |? E (x)| 2 dx = 1. (6.14) Since this integral must be ?nite, (unity in fact), we must have ? E (x)? 0 as x? ±8 in order for the integral to have any hope of converging to a ?nite value. The importance of this with regard to solving the time dependent Schr¨ odinger equation is that we must check whether or not a solution ? E (x) obtained for some chosen value of E satis?es the normalization condition. If it does, then this is a physically acceptable solution, if it does not, then that solution and the corresponding value of the energy are not physically acceptable. The particular case of considerable physical signi?cance is if the potential V (x) is attractive, such as would be found with an electron caught by the attractive Coulomb force of an atomic nucleus, or a particle bound by a simple harmonic potential (a mass on a spring), or, as we have seen in Section 5.3, a particle trapped in an in?nite potential well. In all such cases, we ?nd that except for certain discrete values of the energy, the wave function ? E (x) does not vanish, or even worse, diverges, as x? ±8. In other words, it is only for these discrete values of the energy E that we get physically acceptable wave functions ? E (x), or to put it more bluntly, the particle can never be observed to have any energy other than these particular values, for which reason these energies are often referred to as the â€˜allowedâ€™ energies of the particle. This pairing o? of allowed energy and normalizable wave function is referred to mathematically as ? E (x) being an eigenfunction of the Schr¨ odinger equation, and E the associated energy eigenvalue, a terminology that acquires more meaning when quantum mechanics is looked at from a more advanced standpoint. So we have the amazing result that the probability interpretation of the wave function forces us to conclude that the allowed energies of a particle moving in a potential V (x) are restricted to certain discrete values, these values determined by the nature of the po- tential. This is the phenomenon known as the quantization of energy, a result of quantum mechanics which has enormous signi?cance for determining the structure of atoms, or, to go even further, the properties of matter overall. We have already seen an example of this quantization of energy in our earlier discussion of a particle in an in?ntely deep potential 2 Note that the time dependence has cancelled out because |?(x,t)| 2 = |?E(x)e -iEt/! | 2 = |?E(x)| 2 |e -iEt/! | 2 = |?E(x)| 2 since, for any complex number of the form exp(if), we have |exp(if)| 2 = 1. Chapter 6 The Schr¨ odinger Wave Equation 46 well, though we did not derive the results by solving the Schr¨ odinger equation itself. We will consider how this is done shortly. The requirement that ?(x) ? 0 as x ? ±8 is an example of a boundary condition. Energy quantization is, mathematically speaking, the result of a combined e?ort: that ?(x) be a solution to the time independent Schr¨ odinger equation, and that the solution satisfy these boundary conditions. But both the boundary condition and the Schr¨ odinger equation are derived from, and hence rooted in, the nature of the physical world: we have here an example of the unexpected relevance of purely mathematical ideas in formulating a physical theory. Continuity Conditions There is one additional proviso, which was already mentioned brie?y above, that has to be applied in some cases. If the potential should be discontinuous in some way, e.g. becoming in?nite, as we have seen in the in?nite potential well example, or having a ?nite discontinuity as we will see later in the case of the ?nite potential well, it is possible for the Schr¨ odinger equation to have solutions that themselves are discontinuous. But discontinuous potentials do not occur in nature (this would imply an in?nite force), and as we know that for continuous potentials we always get continuous wave functions, we then place the extra conditions that the wave function and its spatial derivative also must be continuous 3 . We shall see how this extra condition is implemented when we look at the ?nite potential well later. Bound States and Scattering States But what about wave functions such as the harmonic wave function ?(x,t) = Aexp[i(kx-?t)]? These wave functions represent a particle having a de?nite energy E = !? and so would seem to be legitimate and necessary wave functions within the quantum theory. But the problem here, as has been pointed out before in Chapter 5, is that?(x,t) does not vanish as x? ±8, so the normalization condition, Eq. (6.14) cannot be satis?ed. So what is going on here? The answer lies in the fact that there are two kinds of wave functions, those that apply for particles trapped by an attractive potential into what is known as a bound state, and those that apply for particles that are free to travel to in?nity (and beyond), otherwise known as scattering states. A particle trapped in an in?nitely deep potential well is an example of the former: the particle is con?ned to move within a restricted region of space. An electron trapped by the attractive potential due to a positively charged atomic nucleus is also an example â€“ the electron rarely moves a distance more than ~10 nm from the nucleus. A nucleon trapped within a nucleus by attractive nuclear forces is yet another. In all these cases, the probability of ?nding the particle at in?nity is zero. In other words, the wave function for the particle satis?es the boundary condition that it vanish at in?nity. So we see that it is when a particle is trapped, or con?ned to a limited region of space by an attractive potential V (x) (or V (r) in three dimensions), we obtain wave functions that satisfy the above boundary condition, and hand in hand with this, we ?nd that their energies are quantized. But if it should be the case that the particle is free to move as far as it likes in space, in other words, if it is not bound by any attractive potential, (or even repelled by a repulsive potential) then we ?nd that the wave function need not vanish at in?nity, and nor is its energy quantized. The problem of how to reconcile this with the normalization condition, and the probability interpretation of the wave function, is a delicate mathematical issue which we cannot hope to address here, but it can be done. Su?ce to say that provided the wave function does not diverge at in?nity (in 3 The one exception is when the discontinuity is in?nite, as in the case of the in?nite potential well. In that case, only the wave function is reguired to be continuous.Read More

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