1_DoubleSlit - I. The Double Slit Experiment I. The Double...

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Unformatted text preview: I. The Double Slit Experiment I. The Double Slit Experiment x 1. Electron gun or laser x=0 2. (Any QM “particles”) Let P(x) be the probability that an electron will arrive at point x. What will we see? (ignoring single slit diffraction) 2 I. The Double Slit Experiment A. The Classical Picture P(x) = P1(x)+P2(x) P1(x): Probability of passing through slit 1 and reaching position x. P2(x): Probability of passing through slit 2 and reaching position x. Only slit 1 open Only slit 2 open Both slits open + = Bransden and Joachain page 53 3 I. The Double Slit Experiment B. Reality P(x) ≠ P1(x)+P2(x) Pattern characteristic of wave interference This is true even for single electrons let through one at a time Individual electrons detected at specific points (particle-like) but diffraction pattern builds up over time (wave like) Only slit 1 open Only slit 2 open Both slits open + = Bransden and Joachain page 53 4 I. The Double Slit Experiment B. Reality Pattern characteristic of wave interference This is true even for single electrons let through one at a time Individual electrons detected at specific points (particle-like) but diffraction pattern builds up over time (wave like) Bransden and Joachain page 54 5 I. The Double Slit Experiment C. The Quantum Picture For waves, intensity is described by the square of an amplitude. We need something like that now, but something that accounts for the particle aspect. instead of intensity instead of amplitude like a wave, it can be, in general, complex P (r, t) ∝ |ψ (r, t)| = ψ (r, t)ψ (r, t) 2 Probability of detecting a particle at r,t (real positive number) Probability amplitude (wave function, state function) 6 ∗ I. The Double Slit Experiment C. The Quantum Picture P (r, t) ∝ |ψ (r, t)| = ψ (r, t)ψ (r, t) 2 2 Only slit 1 open 2 ∗ Only slit 2 open P1 ∝ |ψ1 | P2 ∝ |ψ2 | P ∝ |ψ1 + ψ2 |2 Both slits open 7 I. The Double Slit Experiment D. More Formally The probability of finding a particle within the volume element dr = dxdydz centered about the point r = (x,y,z) at a time t is P (r, t)dr = |ψ (r, t)| dr 2 so that P (r, t) ∝ |ψ (r, t)|2 = ψ (r, t)ψ ∗ (r, t) Probability density Theory has a statistical nature: one predicts the probability of a result, not a specific result. 8 I. The Double Slit Experiment D. More Formally The particle must be detectable somewhere so the total probability should be 1 (100% chance of finding it somewhere). ￿ P (r, t)dr = ￿ |ψ (r, t)|2 dr = 1 9 I. The Double Slit Experiment E. The Superposition Principle What is the probability density for the double slit experiment? Instead of adding probabilities (classical), one adds wavefunctions. Superposition: If ψ1 describes a possible state of a system, and ψ2 describes another possible state of the system, then any linear combination ψ = c1 ψ1 + c2 ψ2 is also a possible state of the system. constant are, in general, complex too 10 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics We’ve seen two of the postulates of quantum physics. From Bransden and Joachain: 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 11 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. This is also a possible state ψ = c1 ψ1 + c2 ψ2 + · · · one possible state 12 another possible state I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · What are the other postulates? 3. With every dynamical variable is associated a linear operator. 13 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. 14 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. 5. If a series of measurements is made of a dynamical variable A on an ensemble of systems described by the wavefunction ψ, the expectation value, or average value of this dynamical variable is ￿ drψ ∗ (r)Aψ (r) ￿ψ |A|ψ ￿ ￿A￿ = =￿ ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) 15 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. ￿ 5. If a series of measurements is made of a dynamical variable A on an drψ ∗ (r)Aψ (r) ￿ψ |A|ψ ￿ ￿ ￿A￿ = = ensemble of systems described by the wavefunction ψ, the ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable. 16 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. ￿ 5. If a series of measurements is made of a dynamical variable A on an drψ ∗ (r)Aψ (r) ￿ψ |A|ψ ￿ ￿ ￿A￿ = = ensemble of systems described by the wavefunction ψ, the ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger equation ∂ψ (r, t) i￿ = Hψ (r, t) ∂t 17 I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. ￿ 5. If a series of measurements is made of a dynamical variable A on an drψ ∗ (r)Aψ (r) ￿ψ |A|ψ ￿ ￿ ￿A￿ = = ensemble of systems described by the wavefunction ψ, the ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger equation: i￿ ∂ψ (r, t) = Hψ (r, t) ∂t where H is the Hamiltonian, or total energy operator, of the system. 18 a r b ce ea g sp lt A er r a I. The Double Slit Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” 2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · e in L 3. With every dynamical variable is associated a linear operator. lb i H 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A. a n o ￿ 5. If a series of measurements is made of a dynamical variable A on an drψ ∗ (r)Aψ (r) ￿ψ |A|ψ ￿ ￿ ￿A￿ = = ensemble of systems described by the wavefunction ψ, the ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger equation: i￿ ∂ψ (r, t) = Hψ (r, t) ∂t where H is the Hamiltonian, or total energy operator, of the system. 19 We need to learn this formalism of quantum mechanics ...
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This note was uploaded on 09/28/2011 for the course PHYSICS 334 taught by Professor Russel during the Spring '11 term at Waterloo.

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