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Quantum Postulates “Mastery of Fundamentals” Questions CH351 – Prof. Wu Here are some questions to test your mastery of the fundamentals of the postulates of quantum mechanics. Once you’ve mastered the material, you should be able to answer these questions without reference to your notes or textbook. 1. List the postulates of quantum mechanics and explain them. Postulate 1: The state of a system is determined by the wavefunction. Postulate 2: To every classical observable corresponds a linear Hermitian operator in QM. Postulate 3: A measurement of the observable with operator Aˆ can only result in an eigenvalue of that operator. Postulate 4: Given that the system is in state ψ , the expectation value of the * ˆ observable with operator Aˆ is given by A = ∫ ψ€ Aψdx . all space Postulate 5: The wave function evolves in time according to the time-dependent ∂Ψ(x,t)€ ˆ Schroedinger’s € equation ih = HΨ(x,t) . ∂t € 2. What are the properties needed of a wavefunction to be well-behaved? A wavefunction€is well-behaved if it is normalizable, and if it and its first-derivative are single-valued, continuous and finite. 3. Why do we say the wavefunction completely specifies the state of a system? How do we use the wavefunction? We mean that any physically observable quantity is determined, although perhaps probabilistically, by the wavefunction alone. The wavefunction is used according to the postulates above to determine outcomes of measurements. 4. Given a classical observable, write down the corresponding quantum operator. This is done by writing the classical observable in terms of x and p. Then we ∂ substitute the operators xˆ = x and pˆ = −ih for x and p. Note the x operator just ∂x means multiply by x. € possible outcomes of observing a quantity, and how is it related to 5. What are the its quantum operator?€ How do we determine the probabilities associated with each outcome? As stated in postulate 3 above, the outcomes can only be the eigenvalues, an, of the corresponding operator Aˆ , namely Aˆ φ n (x) = an φ n (x) . Given that the system is in a 2 state ψ , the probability of being found in a state j is pn = c n , where the cn are the € € € € Quantum Postulates “Mastery of Fundamentals” Questions CH351 – Prof. Wu coefficients of the expansion of ψ = ∑ c n φ n (x) in terms of a linear combination of the eigenfunctions of Aˆ . n 6. What is an expectation value? How do we determine the expectation value given € a wavefunction? € The expectation value is the average value of an observable over several measurements of identical systems with the same wavefunction. It is determined using postulate 4 above, or by taking an average with a knowledge of the probabilities of each outcome as given by the answer to the preceding question. 7. What does it mean that two wavefunctions are orthogonal to each other? That a set of wavefunctions is orthonormal? Two functions f(x) and g(x) are orthogonal if and only if ∫ f * (x)g(x)dx = 0 , where all space x means the positions (one or more dimensional) available to the system or particle. A set of wavefunctions is orthonormal if and only if any pair of wavefunctions are orthogonal, each each wavefunction is normalized. € 8. When do the eigenfunctions of an operator form an orthonormal basis? The eigenfunctions of a linear Hermitian operator, once normalized, form an orthonormal basis. Thus the eigenfunctions of any operator corresponding to a physical observable form an orthonormal basis. 9. What is a commutator? How is the commutator related to whether two quantities can be observed simultaneously to arbitrary accuracy? The commutator of operator Aˆ and Bˆ is Aˆ , Bˆ = Aˆ Bˆ − Bˆ Aˆ . Two quantities can be [ ] observed simultaneously to arbitrary accuracy if and only if the commutator of their operators is zero (i.e., if they commute). € € € the time-dependent and time-independent 10. What is the difference between Schroedinger’s equation? The time-dependent Schroedinger’s equation is used to determine how a wavefunction changes in time. The time-dependent Schroedinger’s equation, which can be derived from the time-dependent Schroedinger’s equation, is used to find the stationary states of a system. Quantum Postulates “Mastery of Fundamentals” Questions CH351 – Prof. Wu In addition, you should feel comfortable doing problems like those that have been assigned in homework. Here are some additional problems you should feel comfortable doing once you’ve mastered the material. 1. Given a system that consists of two independent degrees of freedom (e.g. two separated particles, two different masses on springs, one mass moving in the x and y directions independently), be able to write down the Hamiltonian. From this, show that the energies are independent and are additive, while the wavefunctions are products of the independent wavefunctions of each individual system. Hˆ = Hˆ x + Hˆ y , where Hˆ x is the Hamiltonian for the system in the x direction (PIB, harmonic oscillator, etc.), and similarly for y. Since Hˆ depends only on x, and x € similarly for y, we can solve the Schroedinger’s equation in each dimension separately, € namely Hˆ xψ n x (x) = E n xψ n x (x) and Hˆ y φ n y (y) = E n y φ n y (y) . From this we can thus show that ψ (x)φ (y) is an eigenfunciton of Hˆ with eigenvalue € nx ny y E nx + E ny : € € ˆ ˆ ˆ Hψ n x (x)φ€n y (y) = H x + H y ψ n x (x)φ n y (y) ( € ) € = Hˆ xψ n x (x)φ n y (y) + Hˆ yψ n x (x)φ n y (y) = φ n y (y) Hˆ xψ n x (x) + ψ n x (x) Hˆ y φ n y (y) ( ) = φ n y (y)E n xψ n x (x) + ψ n x (x)E n y φ n y (y) = E n x + E n y ψ n x (x)φ n y (y) €