Time Development Recall the rectangular initial wave packet in the - PowerPoint PPT Presentation

Time Development Recall the rectangular initial wave packet in the infinite square well shown below. How does it evolve in time? V ( x ) = 0 0 < x < a = ∞ x ≤ 0 and x ≥ a 1 | Ψ( x, 0) � = √ x 0 ≤ x ≤ x 1 and d = x 0 − x 1 d = 0 otherwise V(x) Ψ (x,0) x 0 x 0 a x 1 Experimental Foundations – p. 1/

Probabilities of Different States Square Wave in a Square Well 0.4 0.3 Probability 0.2 0.1 0 500 1000 1500 2000 2500 3000 Energy � eV � Experimental Foundations – p. 2/

Time Development of Square Wave 8 8 t � 0.‘ t � 0.004‘ 6 6 � Ψ � 2 � Ψ � 2 4 4 2 2 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 x x 8 8 t � 0.008‘ t � 0.012‘ 6 6 � Ψ � 2 � Ψ � 2 4 4 2 2 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 x x Experimental Foundations – p. 3/

Time Development of the Initial Gaussian Recall the Gaussian initial wave packet for the free particle shown below. How does it evolve in time? 1 (2 πσ 2 ) 1 / 4 e − x 2 / 4 σ 2 V ( x ) = 0 | Ψ( x, 0) � = 0.5 0.4 0.3 Y 0.2 0.1 � 3 � 2 � 1 0 1 2 3 4 x Experimental Foundations – p. 4/

Time Development of the Initial Gaussian t � 0 t � 1.5 5 5 4 4 3 3 � Ψ � 2 P 2 2 1 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 x x t � 3. t � 4.5 5 5 4 4 3 3 P P 2 2 1 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 x x Experimental Foundations – p. 5/

Time Development of Nuclear Fusion Consider a case of one dimensional nuclear ‘fusion’. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at x = 0 and x = L . The eigenfunctions and eigenvalues are � E n = n 2 � 2 π 2 2 � nπx � 0 ≤ x ≤ a φ n = a sin 2 ma 2 a = 0 x < 0 and x > a . The neutron is in the n = 4 state when it fuses with another nucleus that is the same size, instantly putting the neutron in a new infinite square well with walls at x = 0 and x = 2 a . 1. What are the new eigenfunctions and eigenvalues of the fused system? 2. How will the initial wave packet evolve in time? Experimental Foundations – p. 6/

Time Development of Nuclear Fusion t � 0. t � 0.3 0.2 0.2 0.15 0.15 Ψ�Ψ Ψ�Ψ 0.1 0.1 0.05 0.05 2.5 5 7.5 10 12.5 15 17.5 20 2.5 5 7.5 10 12.5 15 17.5 20 x � fm � x � fm � t � 0.6 t � 0.9 0.2 0.2 0.15 0.15 Ψ�Ψ Ψ�Ψ 0.1 0.1 0.05 0.05 2.5 5 7.5 10 12.5 15 17.5 20 2.5 5 7.5 10 12.5 15 17.5 20 x � fm � x � fm � Experimental Foundations – p. 7/