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Quantum One: Lecture 3

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Implications of Schrödinger's Wave Mechanics for Conservative Systems

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In the last lecture we stated the postulates of Schrödinger’s wave mechanics

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We also extended the notation as written in the postulates to include observables with a continuous spectrum In the process we deduced the spectrum and eigenfunctions of the position operator:

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We also extended the notation as written in the postulates to include observables with a continuous spectrum In the process we deduced the spectrum and eigenfunctions of the position operator:

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In this lecture we want to explore some of the general implications of Schrödinger’s postulates as they apply to conservative systems, that is to particles for which the Hamiltonian operator is independent of time. You probably recall, that for many physical systems, one is interested in solving the so-called initial value problem: Given the dynamical state of the system at some initial time t = 0, find the state into which it evolves at an arbitrary time later. That is, given an arbitrary initial wave function find the wave function into which it evolves under Schrödinger’s equation of motion.

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In this lecture we want to explore some of the general implications of Schrödinger’s postulates as they apply to conservative systems, that is to particles for which the Hamiltonian operator is independent of time. You probably recall, that for many physical systems, one is interested in solving the so-called initial value problem: Given the dynamical state of the system at some initial time t = 0, find the state into which it evolves at an arbitrary time later. That is, given an arbitrary initial wave function find the wave function into which it evolves under Schrödinger’s equation of motion.

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In this lecture we want to explore some of the general implications of Schrödinger’s postulates as they apply to conservative systems, that is to particles for which the Hamiltonian operator is independent of time. You probably recall, that for many physical systems, one is interested in solving the so-called initial value problem: Given the dynamical state of the system at some initial time t = 0, find the state into which it evolves at an arbitrary time later. That is, given an arbitrary initial wave function find the wave function into which it evolves under Schrödinger’s equation of motion.

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For a particle with a time-independent Hamiltonian, the initial value problem can in principle be solved, using the process of separation of variables Consider: When the scalar potential energy field that the particle moves under is independent of time (so that ∂V/∂t=0) then so is the total Hamiltonian In this case it is possible to solve the equation of motion by first looking for special, separable solutions of the form

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For a particle with a time-independent Hamiltonian, the initial value problem can in principle be solved, using the process of separation of variables Consider: When the scalar potential energy field that the particle moves under is independent of time (so that ∂V/∂t=0) then so is the total Hamiltonian In this case it is possible to solve the equation of motion by first looking for special, separable solutions of the form

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For a particle with a time-independent Hamiltonian, the initial value problem can in principle be solved, using the process of separation of variables Consider: When the scalar potential energy field that the particle moves under is independent of time (so that ∂V/∂t=0) then so is the total Hamiltonian In this case it is possible to solve the equation of motion by first looking for special, separable solutions of the form

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Substitution into Schrödinger's equation then gives for a separable solution: Dividing by the same product ψ=φχ one obtains To be equal at all times and positions, they must both equal a constant. Introduce a separation constant E (having units of energy);

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Substitution into Schrödinger's equation then gives for a separable solution: Dividing by the same product ψ=φχ one obtains To be equal at all times and positions, they must both equal a constant. Introduce a separation constant E (having units of energy);

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Substitution into Schrödinger's equation then gives for a separable soloution: Dividing by the same product ψ=φχ one obtains To be equal at all times and positions, they must both equal a constant. We thus introduce a separation constant E (having units of energy);

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Substitution into Schrödinger's equation then gives for a separable soloution: Dividing by the same product ψ=φχ one obtains To be equal at all times and positions, they must both equal a constant. We thus introduce a separation constant E (having units of energy);

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This gives two separated differential equations: The time-dependence of the function χ(t) is governed by an easily integrable first-order equation: It is convenient to set χ(0)=1, letting φ(r) absorb any multiplicative normalization constants.

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This gives two separated differential equations: The time-dependence of the function χ(t) is governed by an easily integrable first-order equation: It is convenient to set χ(0)=1, letting φ(r) absorb any multiplicative normalization constants.

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This gives two separated differential equations: The time-dependence of the function χ(t) is governed by an easily integrable first-order equation: It is convenient to set χ(0)=1, letting the spatial part φ(r) absorb any multiplicative normalization constants.

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Thus, separable solutions of the form will exist, provided that there are acceptable spatial functions that satisfies the 2 nd order partial differential equation:

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Comment 1: This equation obeyed by the spatial part of the wave function is clearly just the eigenvalue equation for the Hamiltonian operator H representing the total energy of the system. It is the energy eigenvalue equation. [Note: Some author's refer to this last equation also as "Schrödinger's equation", or the "time-independent Schrödinger equation". To avoid confusion, we shall reserve the term "Schrödinger equation" exclusively for the evolution equation appearing in the postulates, and simply refer to this equation as the "energy eigenvalue equation".]

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Comment 1: This equation obeyed by the spatial part of the wave function is clearly just the eigenvalue equation for the Hamiltonian operator H representing the total energy of the system. It is the energy eigenvalue equation. [Note: Some author's refer to this last equation also as "Schrödinger's equation", or the "time-independent Schrödinger equation". To avoid confusion, we shall reserve the term "Schrödinger equation" exclusively for the evolution equation appearing in the postulates, and simply refer to this equation as the "energy eigenvalue equation".]

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Comment 2: In solving eigenvalue equations of this sort, physical conditions will restrict the values of E for which physically acceptable solutions exist. Such regularity conditions include, e.g., normalizeability, continuity, differentiability, and the requirement that there be finite energy content in any finite region of space. In general, one seeks solutions that that are as continuous as possible while still solving the underlying differential equation. The values of E for which acceptable solutions exist we identify with the energy eigenvalues of the system. These are the only values that can be obtained in an energy measurement. The eigenfunctions are the states the system can be left in during an energy measurement.

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Comment 2: In solving eigenvalue equations of this sort, physical conditions will restrict the values of E for which physically acceptable solutions exist. Such regularity conditions include, e.g., normalizeability, continuity, differentiability, and the requirement that there be finite energy content in any finite region of space. In general, one seeks solutions that that are as continuous as possible while still solving the underlying differential equation. The values of E for which acceptable solutions exist we identify with the energy eigenvalues of the system. These are the only values that can be obtained in an energy measurement. The eigenfunctions are the states the system can be left in during an energy measurement.

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Comment 2: In solving eigenvalue equations of this sort, physical conditions will restrict the values of E for which physically acceptable solutions exist. Such regularity conditions include, e.g., normalizeability, continuity, differentiability, and the requirement that there be finite energy content in any finite region of space. In general, one seeks solutions that that are as continuous as possible while still solving the underlying differential equation. The values of E for which acceptable solutions exist we identify with the energy eigenvalues of the system. These are the only values that can be obtained in an energy measurement. The eigenfunctions are the states the system can be left in during an energy measurement.

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Comment 2 : As we have suggested the issue of normalizability is related to the nature of the spectrum of an observable. For a particle moving in a potential energy field V that goes to zero at infinity, there are generally two type of solutions: 1. Bound state solutions, found at negative energy, in which the particle is localized around a force center. Such solutions are usually associated with discrete, quantized energy eigenvalues and are square normalizeable. In this case the energy eigenfunctions satisfy the boundary condition

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Comment 2 : As we have suggested the issue of normalizability is related to the nature of the spectrum of an observable. For a particle moving in a potential energy field V that goes to zero at infinity, there are generally two type of solutions: 1. Bound state solutions, found at negative energy, in which the particle is localized around a force center. Such solutions are usually associated with discrete, quantized energy eigenvalues and are square normalizeable. In this case the energy eigenfunctions satisfy the boundary condition

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Comment 2: 2. Continuum or scattering solutions, often found at positive energy, in which the particle is delocalized, and is not bound to the force center (but may describe scattering). Such solutions are usually associated with continuous energy eigenvalues and are not square normalizeable. In this case the energy eigenfunctions simply have to remain bounded, i.e., we require that there exist a constant C such that In solving the eigenvalue equation we look for both types of acceptable solutions.

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Comment 2: 2. Continuum or scattering solutions, often found at positive energy, in which the particle is delocalized, and is not bound to the force center (but may describe scattering). Such solutions are usually associated with continuous energy eigenvalues and are not square normalizeable. In this case the energy eigenfunctions simply have to remain bounded, i.e., we require that there exist a constant C such that In solving the eigenvalue equation we look for both types of acceptable solutions in order to obtain a maximally complete set.

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Comment 3: Once we have solved the energy eigenvalue equations, then we implicitly have a limited set of separable solutions to the basic initial value problem: If the initial state of the system happens to be one of the energy eigenstates we found, so that then for t > 0, the system stays in a spatial eigenstate of H, it just acquires a time- dependent phase factor.

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Comment 3: Once we have solved the energy eigenvalue equations, then we implicitly have a limited set of separable solutions to the basic initial value problem: If the initial state of the system happens to be one of the energy eigenstates we found, so that then for t > 0, the system stays in a spatial eigenstate of H, it just acquires a time- dependent phase factor.

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Comment 3: Once we have solved the energy eigenvalue equations, then we implicitly have a limited set of separable solutions to the basic initial value problem: If the initial state of the system happens to be one of the energy eigenstates we found, so that then for t > 0, the system stays in a spatial eigenstate of H, it just acquires a time- dependent phase factor.

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Comment 3: Although these special separable solutions have a time-dependent phase factor, they are often called stationary states, because the probability density associated with the particle’s position does not change in time, the particle does not move. For this initial condition In fact, one can show that in such a stationary state, no physical properties of the particle change in time. Energy eigenstates are important, but dynamically boring!

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Comment 3: Although these special separable solutions have a time-dependent phase factor, they are often called stationary states, because the probability density associated with the particle’s position does not change in time, the particle does not move. For this initial condition In fact, one can show that in such a stationary state, no physical properties of the particle change in time. Energy eigenstates are important, but dynamically boring!

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Comment 3: Although these special separable solutions have a time-dependent phase factor, they are often called stationary states, because the probability density associated with the particle’s position does not change in time, the particle does not move. For this initial condition In fact, one can show that in such a stationary state, no physical properties of the particle change in time. Energy eigenstates are important, but dynamically boring!

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Q: So have we solved the initial value problem? A: Not yet! only for the very rare case when the initial state is an energy eigenstate of the Hamiltonian! Q: So what do we do when the initial state is not an energy eigenstate? A: We go back to the postulates! And we note that once we have solved the energy eigenvalue problem, we will have determined the eigenfunctions of H. Suppose H has a discrete spectrum and we enumerate our solutions so that

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Q: So have we solved the initial value problem? A: Not yet! only for the very rare case when the initial state is an energy eigenstate of the Hamiltonian! Q: So what do we do when the initial state is not an energy eigenstate? A: We go back to the postulates! And we note that once we have solved the energy eigenvalue problem, we will have determined the eigenfunctions of H. Suppose H has a discrete spectrum and we enumerate our solutions so that

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Q: So have we solved the initial value problem? A: Not yet! only for the very rare case when the initial state is an energy eigenstate of the Hamiltonian! Q: So what do we do when the initial state is not an energy eigenstate? A: We go back to the postulates! And we note that once we have solved the energy eigenvalue problem, we will have determined the eigenfunctions of H. Suppose H has a discrete spectrum and we enumerate our solutions so that

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Q: So have we solved the initial value problem? A: Not yet! only for the very rare case when the initial state is an energy eigenstate of the Hamiltonian! Q: So what do we do when the initial state is not an energy eigenstate? A: We go back to the postulates! And we note that once we have solved the energy eigenvalue problem, we will have determined the eigenfunctions of H. Suppose H has a discrete spectrum and we enumerate our solutions so that

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Q: So have we solved the initial value problem? A: Not yet! only for the very rare case when the initial state is an energy eigenstate of the Hamiltonian! Q: So what do we do when the initial state is not an energy eigenstate? A: We go back to the postulates! And we note that once we have solved the energy eigenvalue problem, we will have determined the eigenfunctions of H. Suppose H has a discrete spectrum and we enumerate our solutions so that

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But energy is an observable! Energy eigenstates are sufficiently complete that an arbitrary initial state can be expanded in energy eigenfunctions: And we know how each energy eigenfunction evolves. Q: Does that helps us in a linear superposition of energy eigenstates? A: Yes! Because Schrödinger's equation of motion is both First order in time and Linear

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But energy is an observable! Energy eigenstates are sufficiently complete that an arbitrary initial state can be expanded in energy eigenfunctions: And we know how each energy eigenfunction evolves. Q: Does that helps us in a linear superposition of energy eigenstates? A: Yes! Because Schrödinger's equation of motion is both First order in time and Linear

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But energy is an observable! Energy eigenstates are sufficiently complete that an arbitrary initial state can be expanded in energy eigenfunctions: And we know how each energy eigenfunction evolves. Q: Does that helps us in a linear superposition of energy eigenstates? A: Yes! Because Schrödinger's equation of motion is both First order in time and Linear

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But energy is an observable! Energy eigenstates are sufficiently complete that an arbitrary initial state can be expanded in energy eigenfunctions: And we know how each energy eigenfunction evolves. Q: Does that helps us in a linear superposition of energy eigenstates? A: Yes! Because Schrödinger's equation of motion is both First order in time and Linear

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First order in time, means that the final state is uniquely determined by the initial state: Linearity, means that any linear superposition of solutions to the time- dependent equation is itself a solution. Together these imply that If then

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First order in time, means that the final state is uniquely determined by the initial state: Linearity, means that any linear superposition of solutions to the time- dependent equation is itself a solution. Together these imply that If then

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First order in time, means that the final state is uniquely determined by the initial state: Linearity, means that any linear superposition of solutions to the time- dependent equation is itself a solution. Together these imply that If then

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First order in time, means that the final state is uniquely determined by the initial state: Linearity, means that any linear superposition of solutions to the time- dependent equation is itself a solution. Together these imply that If then

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Applying this idea to our (complete set) of separable solutions, we deduce that if then under Schrödinger's equation:

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Applying this idea to our (complete set) of separable solutions, we deduce that if then under Schrödinger's equation:

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Applying this idea to our (complete set) of separable solutions, we deduce that if then under Schrödinger's equation:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following 3 step prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following 3 step prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following 3 step prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following 3 step prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Q: OK, so have we solved the initial value problem, yet? A: Yes! (In principle….) We have the following 3 step prescription: Given: 1) The Hamiltonian H = T + V (i.e., given V, independent of time) 2) An arbitrary initial state To find: The state at a later time t > 0 1)Solve: 2)Determine: the initial amplitudes λ n in the expansion 3)Evolve:

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Comments: Obviously, we have avoided some technical details such as 1.How do we actually solve the energy eigenvalue problem? 2.How do we determine the amplitudes that allow us to expand an arbitrary state in energy eigenfunctions? We will address each of these more fully in time, as we get a few more definitions in our toolbox. In the meantime we make a few more comments

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Comments: Obviously, we have avoided some technical details such as 1.How do we actually solve the energy eigenvalue problem? 2.How do we determine the amplitudes that allow us to expand an arbitrary state in energy eigenfunctions? We will address each of these more fully in time, as we get a few more definitions in our toolbox. In the meantime we make a few more comments

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Comments: Obviously, we have avoided some technical details such as 1.How do we actually solve the energy eigenvalue problem? 2.How do we determine the amplitudes that allow us to expand an arbitrary state in energy eigenfunctions? We will address each of these more fully in time, as we get a few more definitions in our toolbox. In the meantime we make a few more comments

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Comments: Obviously, we have avoided some technical details such as 1.How do we actually solve the energy eigenvalue problem? 2.How do we determine the amplitudes that allow us to expand an arbitrary state in energy eigenfunctions? We will address each of these more fully in time, as we get a few more definitions in our toolbox. In the meantime we make a few more comments

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Comments: 1.We note first that the general time dependent solution to the Schrödinger equation is not stationary. (Thank goodness!) 2.In a linear superposition of eigenfunctions with different energies, there is not a single phase factor that you can factor out of the whole expression. The energy E is just not well-defined, it doesn’t exist, so

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Comments: 1.We note first that the general time dependent solution to the Schrödinger equation is not stationary. (Thank goodness!) 2.In a linear superposition of eigenfunctions with different energies, there is not a single phase factor that you can factor out of the whole expression. The energy E is just not well-defined, it doesn’t exist, so

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Comments: 1.We note first that the general time dependent solution to the Schrödinger equation is not stationary. (Thank goodness!) 2.In a linear superposition of eigenfunctions with different energies, there is not a single phase factor that you can factor out of the whole expression. The energy E is just not well-defined, it doesn’t exist, so

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Comments: This is a hard notion to drive from the imagination of many students, so I repeat, with emphasis, in GENERAL Many students look at this and read the very clear not equal sign as an equal sign. Stop. Pay attention. I will never give a problem in this course in which the answer is the equation given above with the not equal sign replaced by an equal sign. I have taken 1000 pts off on papers where such an equation has been given as the answer.

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Comments: This is a hard notion to drive from the imagination of many students, so I repeat, with emphasis, in GENERAL Many students look at this and read the very clear not equal sign as an equal sign. Stop. Pay attention. I will never give a problem in this course in which the answer is the equation given above with the not equal sign replaced by an equal sign. I have taken 1000 pts off on papers where such an equation has been given as the answer.

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Comments: This is a hard notion to drive from the imagination of many students, so I repeat, with emphasis, in GENERAL Many students look at this and read the very clear not equal sign as an equal sign. Stop. Pay attention. I will never give a problem in this course in which the answer is the equation given above with the not equal sign replaced by an equal sign. I have taken 1000 pts off on papers where such an equation has been given as the answer.

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Comments: 3.Next we note that the general solution can be put in the suggestive form where Thus, the wave function can be expanded at each instant t in energy eigenfunctions, in terms of amplitudes defined at that instant of time. 4.But if we know the eigenfunctions, this suggests an alternative way of specifying or representing the dynamical state

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Comments: 3.Next we note that the general solution can be put in the suggestive form where Thus, the wave function can be expanded at each instant t in energy eigenfunctions, in terms of amplitudes defined at that instant of time. 4.But if we know the eigenfunctions, this suggests an alternative way of specifying or representing the dynamical state

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Comments: 3.Next we note that the general solution can be put in the suggestive form where Thus, the wave function can be expanded at each instant t in energy eigenfunctions, in terms of amplitudes defined at that instant of time. 4.But if we know the eigenfunctions, this suggests an alternative way of specifying or representing the dynamical state

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Comments: 5.Schrodinger tells us: the dynamical state is represented by the wave function which in principle means specifying a different complex number at each point in space. We have to (somehow) specify an infinite set of complex numbers. 6.But if you and I both know the energy eigenfunctions, than we can also, at any instant, tell each other what the state is by simply giving the compete set {λ n } of complex expansion coefficients. We could order them in a column vector for example, and use that column vector to represent the state, just as we do with the position vectors of particles, which we can represent by a row vector (x,y,z). 7.The idea here is that there may be more than one equivalent way of specifying the underlying dynamical state of the quantum system besides that given by Schrödinger's postulate.

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Comments: 5.Schrodinger tells us: the dynamical state is represented by the wave function which in principle means specifying a different complex number at each point in space. We have to (somehow) specify an infinite set of complex numbers. 6.But if you and I both know the energy eigenfunctions, than we can also, at any instant, tell each other what the state is by simply giving the compete set {λ n } of complex expansion coefficients. We could order them in a column vector for example, and use that column vector to represent the state, just as we do with the position vectors of particles, which we can represent by a row vector (x,y,z).

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Comments: 5.Schrodinger tells us: the dynamical state is represented by the wave function which in principle means specifying a different complex number at each point in space. We have to (somehow) specify an infinite set of complex numbers. 6.But if you and I both know the energy eigenfunctions, than we can also, at any instant, tell each other what the state is by simply giving the compete set {λ n } of complex expansion coefficients. We could order them in a column vector for example, and use that column vector to represent the state, just as we do with the position vectors of particles, which we can represent by a column or a row vector (x,y,z).

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Comments: The idea here is that there may be more than one equivalent way of specifying the underlying dynamical state of the quantum system besides that given by Schrödinger's postulate. This will be useful to us in motivating the general formulation of quantum mechanics as it applies to arbitrary quantum mechanical systems. Before we do that, though, we apply what we’ve learned about the evolution of conservative systems, to a concrete and seemingly simply, but extremely important example: The Free Particle Which forms the focus of the next lecture.

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Comments: The idea here is that there may be more than one equivalent way of specifying the underlying dynamical state of the quantum system besides that given by Schrödinger's postulate. This will be useful to us in motivating the general formulation of quantum mechanics as it applies to arbitrary quantum mechanical systems. Before we do that, though, we apply what we’ve learned about the evolution of conservative systems, to a concrete and seemingly simply, but extremely important example: The Free Particle Which forms the focus of the next lecture.

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Comments: The idea here is that there may be more than one equivalent way of specifying the underlying dynamical state of the quantum system besides that given by Schrödinger's postulate. This will be useful to us in motivating the general formulation of quantum mechanics as it applies to arbitrary quantum mechanical systems. Before we do that, though, we apply what we’ve learned about the evolution of conservative systems, to a concrete and seemingly simply, but extremely important example: The Free Particle Which forms the focus of the next lecture.

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Comments: The idea here is that there may be more than one equivalent way of specifying the underlying dynamical state of the quantum system besides that given by Schrödinger's postulate. This will be useful to us in motivating the general formulation of quantum mechanics as it applies to arbitrary quantum mechanical systems. Before we do that, though, we apply what we’ve learned about the evolution of conservative systems, to a concrete and seemingly simply, but extremely important example: The Free Particle Which forms the focus of the next lecture.

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