Chapter 4 Unitary Operators and Quantum Dynamics In the previous section, I explained how one can dig out experimentally relevant information using states of a quantum system and operators representing the observ- ables. The remaining burning question, however, is how can we find these states so that we could use these methods. In a typical experiment, an experimentalist begins by “preparing” a quantum system in some state, which they believe they know.1 After that they smash the system with a hammer, or hit it by a laser light, or subject it to an electric or magnetic field, wait for some time, and measure new values of the selected observables. In order to predict the results of new measurements, you must be able to describe how the quantum system changes between the time of preparation and the time of subsequent measurement, or, speaking more scientifically, you must know its dynamics. As it has been made clear in the previous section, you need two objects to predict the results of a measurement: a state of the system and the operator assigned to the measured observable. Now you can ask an interesting question: “When the quantum system evolves in time, what is actually changing—the state or the operator?” To make this question more specific, consider an expectation value of an observable described by operator TO : h˛j TO j˛i. When your system evolves, this expectation value becomes a function of time. The question is, which element of the expression for the expectation value, TO or j˛i, must be considered as a time-dependent quantity to describe the dynamics of the expectation value? It turns out that time dependence can be ascribed to either of these two elements, and depending on the choice, it will generate two different but equivalent pictures of quantum mechanics. In the so-called Schrödinger picture, the state vectors are treated as time-dependent quantities, while operators remain fixed rules transforming the states. In the Heisenberg picture, the state vector is considered as a constant, and all the dynamics of the system is ascribed to the time- dependent operators. The origins of these two pictures can be found in the earlier 1Preparation of a quantum system in a predefined state usually consists in carrying out a measurement, but it is not an easy task to prepare a system in a state we want. © Springer International Publishing AG, part of Springer Nature 2018 95 L.I. Deych, Advanced Undergraduate Quantum Mechanics, https://doi.org/10.1007/978-3-319-71550-6_4 96 4 Unitary Operators and Quantum Dynamics days of quantum theory with the Heisenberg matrix mechanics competing against Schrödinger’s matter wave theory. The first attempt to prove equivalence of the two pictures was undertaken by Schrödinger as early as in 1926, but the rigorous math- ematical proof of the equivalence did not exist until John von Neumann published in 1932 his definitive book Mathematical Foundations of Quantum Mechanics. Von Neumann was one of the major figures in mathematics and mathematical physics of the twentieth century. Born to a rich Jewish family in Hungary, which was elevated to nobility by Austro-Hungary Emperor Franz Joseph (hence the prefix von in his name), he was a child prodigy, got his Ph.D. in mathematics at the age of 23, and became the youngest privatdocent at the University of Berlin. In 1929 he got an offer from Princeton University and moved to the USA. He brought his entire family to America in 1938 saving them from certain death. In addition to laying rigorous mathematical foundation to quantum theory, von Neumann is famous for his role in the Manhattan Project and developing the concept of digital computers (among other things). After this brief historical detour, I begin presentation of quantum dynamics starting with the Schrödinger picture. 4.1 Schrödinger Picture 4.1.1 Time-Evolution Operator and Schrödinger Equation The statistical interpretation of quantum mechanical formalism makes sense only if all vectors describing states of quantum system remain normalized at all times. I will begin digging deeper into this issue by computing the norm of a generic vector k˛k using Eq. 3.39. First, I need the corresponding bra vector: X ˛ h j D an h nj n so that I can write for the norm X X X X X ˛ 2 ˛ ˛ ı 2 : k k D h j i D aman h nj mi D aman nm D janj m n m n n (4.1) 2 According to the postulate 4 in Sect. 3.3.5, janj is equal to probability pn that the respective eigenvalue will be observed. Equation 4.1 in this case can be interpreted as a statement that the norm of a generic vector is equal to the sum of probabilities Pof all possible measurement outcomes. The latter must obviously be equal to unity 1 ˛ n pn D regardless of the time dependence of state j i. This result has quite a profound consequence. Indeed, time dependence of a state vector can be considered as a transformation of a vector j˛ .t0/i defined at some initial instant of time t0 into another vector j˛ .t/i at time t under the action of an operator: 4.1 Schrödinger Picture 97 j˛ .t/i D UO .t; t0/ j˛ .t0/i : (4.2) In order to keep the norm of the vector unchanged, the operator UO .t; t0/ must be unitary, which significantly limits the class of operators that can be used to describe the dynamics of quantum states. It also must obey an obvious condition: UO .t0; t0/ D IO: (4.3) Now, considerˇ ˛ an evolution of the system from state j˛ .t0/i to state j˛ .t1/i and then ˇ to state ˛ tf , which can be described as j˛ .t1/i D UO .t1; t0/ j˛ .t0/i ˇ ˛ ˇ ˛ tf D UO tf ; t1 j˛ .t1/i : I can also describe a system’s dynamics from the initial state to the final, bypassing the intermediate state: ˇ ˛ ˇ ˛ tf D UO tf ; t0 j˛ .t0/i : Comparing this with the first two lines of the previous equation, you can infer an important property of the time-evolution operator UO : UO tf ; t0 D UO tf ; t1 UO .t1; t0/ : (4.4) An important corollary of Eq. 4.4 is obtained by setting tf D t0, which yields 1 UO .t0; t1/ UO .t1; t0/ D IO ) UO .t0; t1/ D UO .t1; t0/ (4.5) where I also used Eq. 4.3. In other words, the reversal of time in quantum dynamics is equivalent to replacing the time-evolution operator with its inverse. This idea can also be expressed by saying that by inverting the time-evolution operator, you describe the evolution of the system from present to the past. This property can also be described as reversibility of quantum dynamics: taking a system from t0 to tf and back brings the system in its original state completely reversing its initial evolution. Now, let me consider the action of UO .t1; t0/ over an infinitesimally small time interval t0; t1 D t0 C dt. Expanding this operator over the small interval dt and using Eq. 4.3, I can write: UO .t0 C dt; t0/ D IO C GdtO (4.6) ˇ ˇ where GO Á dUO =dtˇ is an operator obtained by differentiating the time-evolution tDt0 operator with respect to time. Inverse to the operator defined by Eq. 4.6 can be found 98 4 Unitary Operators and Quantum Dynamics 1 by expanding function .1 C x/ with respect to x and keeping only linear in x terms: Á1 1 .1 C x/ ' 1 x. Applying this to operator IO C GdtO , I get 1 UO .t0 C dt; t0/ D IO GdtO : At the same time, Hermitian conjugation of Eq. 4.6 returns UO .t0 C dt; t0/ D IO C GO dt: Since the time-evolution operator is unitary (UO 1 D UO ), operator GO has to be anti- Hermitian: GO DGO , so that it can be presented as GO DiHO =„ (see Eq. 3.30), where HO is a Hermitian operator and „ is introduced to ensure that HO has the dimension of energy. Indeed, since the time-evolution operator is dimensionless, it is clear that operator GO has the dimension of inverse time. The dimension of the Planck’s constant is that of energy multiplied by time, so it is clear that HO has indeed the dimension of energy. This simple analysis leads the way to the next postulate of quantum theory. Postulate 6 Hermitian operator HO in the expansion of the time-evolution operator is the operator version of Hamiltonian function of classical mechanics. Thus, Eq. 4.6 can now be rewritten as HO UO .t0 C dt; t0/ D IO i dt: (4.7) „ Taking advantage of the composition rule, Eq. 4.4, I can write: UO .t C dt; t0/ D UO .t C dt; t/ UO .t; t0/ D ! HO HO IO i dt UO .t; t0/ D UO .t; t0/ i UO .t; t0/ dt „ „ where I also used Eq. 4.7. The main difference between this last expression and Eq. 4.7 is that t in the latter can be separated from t0 by a finite interval. The last equation can be rewritten in the form of differential equation: dUO .t; t0/ HO Di UO .t; t0/ : (4.8) dt „ Applying Eq. 4.7 to Eq. 4.2, I can also derive: Â Ã i j˛ .t C dt/i j˛ .t/i i j˛ .t C dt/i D IO HdtO j˛ .t/i ) D HO j˛ .t/i „ dt „ 4.1 Schrödinger Picture 99 which can be rewritten in a standard form d j˛i i„ D HO j˛i (4.9) dt called Schrödinger equation.